\documentclass[fleqn]{beamer}
%\documentclass[handout]{beamer}
%\usepackage{pgfpages}
%\pgfpagelayout{2 on 1}[letterpaper, border shrink=5mm]
\mode{\setbeamercolor{background canvas}{bg=black!3}}
\usepackage{epstopdf}
% %% Jose Antonio Abell Mena provided this for DSL descriptions
% % (used in a file _Chapter_SoftwareHardware_Domain_Specific_Language_English.tex
% % This is added for listing FEI DSL
% % since he customized it, it needs to be changed (linked to
% % /usr/share/texmf/tex/latex/misc)
% %\usepackage{myListings}
% \input{essi_listings_options.tex}
\usepackage[absolute,overlay]{textpos}
% for tikzpicture
\usepackage{tikz}
\usetikzlibrary{tikzmark,calc}
\usepackage{threeparttable}
%\usepackage{MnSymbol}%
% for smileys
\usepackage{wasysym}%
\usepackage{multirow}
% for subfloat figures:
\usepackage{subfig}
\usepackage{tabularx}
%used for truncated section title in headers
\usepackage[breakall]{truncate}
%\usepackage{caption}
\usepackage{ulem}
\usepackage{amsmath}
\usepackage{amsmath}
\usepackage{amssymb}
\newcommand\hmmax{0} % default 3
% solution from https://texfaq.org/FAQ-manymathalph
\newcommand{\bmmax}{3}
% \newcommand\bmmax{0} % default 4
\usepackage{bm}
\usepackage{IEEEtrantools}
\usepackage{setspace}
\usepackage{scalerel}
% for listing DSL
\input{essi_listings_options.tex}
\usepackage{multimedia}
% for inclusion of other PDF pages, in this case Frank's presentation
\usepackage{pdfpages}
%This is a macro to convert eps to pdf files on the fly.
% make sure figure syntax uses graphicx syntax NOT epsfig syntax
%from http://mailman.mit.edu/pipermail/macpartners/2005-January/000780.html
%
% \ifx\pdfoutput\undefined
% % we are running LaTeX, not pdflatex
% \usepackage{graphicx}
% \else
% % we are running pdflatex, so convert .pdf files to .pdf
% \usepackage[pdftex]{graphicx}
% \usepackage{epstopdf}
% \fi
% %*****************************************
%% ovo je za cirilicu
\input vuk.def
\newfont{\cyr}{wncyr10 scaled 1000 }
\newfont{\cyb}{wncyb10 scaled 1000 }
\newfont{\cyi}{wncyi10 scaled 1000 }
\newfont{\cysc}{wncysc10 scaled 1000 }
\newfont{\cyss}{wncyss10 scaled 1200 }
\newfont{\romjba}{cmss10 scaled 1000}
\newfont{\cyrjbnaslov}{wncyss10 scaled 1200 }
% % for not showing eq numbers unless eq is references
% \usepackage[fleqn,tbtags]{mathtools}
% \mathtoolsset{
% showonlyrefs
% }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FEI LOGO definition
\newcommand{\FEI}
{{\sf \uppercase{F}\kern-0.20em\uppercase{E}\kern-0.20em\uppercase{I}}}
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % % ovo je za cirilicu
% % \input vuk.def
% % %\newfont{\cyr}{wncyr10 scaled 1200}
% % %\newfont{\cyrimena}{wncyb10 scaled 1800}
% % %\newfont{\cyrnaslov}{wncyb10 scaled 2600}
% % %\newfont{\cyrpodnaslov}{wncyr10 scaled 1400}
%
% \newfont{\cyr}{wncyss10 scaled 1200}
% \newfont{\cyrimena}{wncyss10 scaled 2000}
% \newfont{\cyrnaslov}{wncyss10 scaled 2800}
% \newfont{\cyrpodnaslov}{wncyss10 scaled 1400}
%
% \newfont{\cyn}{wncyss10 scaled 2200}
% %% JB sig
% \newfont{\cyrjb}{wncyr10 scaled 600}
% \newcommand{\JB}
% {{\cyrjb \lower0.50ex\hbox{\uppercase{J}}\kern-.58em\hbox{\uppercase{B}}}}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setbeamertemplate{navigation symbols}{}
\mode
{
% \usetheme{Marburg} % ima naslov i sadrzaj sa desne strane
% \usetheme{Hannover} % ima naslov i sadrzaj sa leve strane
% \usetheme{Singapore} % ima sadrzaj i tackice gore
% \usetheme{Antibes} % ima sadrzaj gore i kao graf ...
% \usetheme{Berkeley} % ima sadrzaj desno
% \usetheme{Berlin} % ima sadrzaj gore i tackice
% \usetheme{Goettingen} % ima sadrzxaj za desne strane
% \usetheme{Montpellier} % ima graf sadrzaj gore
% \usetheme{Warsaw}
% \usetheme{Warsaw}
\usetheme{Dresden}
\usecolortheme[RGB={20,0,128}]{structure}
% or ...
\setbeamercovered{transparent}
% \setbeamercovered{transparent}
% or whatever (possibly just delete it)
% \usecolortheme{albatross} % teget sa svetlim slovima
% \usecolortheme{beetle} % siva pozadina (vrh plav)
\usecolortheme{seagull} % sivo
%%%%%%%
% \usecolortheme{BorisJeremic}
%%%%%%%
% \usecolortheme{rose}
% \usefonttheme[onlylarge]{structuresmallcapsserif}
% \usefonttheme{structuresmallcapsserif}
}
\definecolor{mycolor}{rgb}{0,0.08,0.45}%
\usepackage[greek,english]{babel}
\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\newcommand{\ud}{{\rm d}}
\usepackage{array}
%%%% HYPERREF HYPERREF HYPERREF HYPERREF HYPERREF
%%%% HYPERREF HYPERREF HYPERREF HYPERREF HYPERREF
\definecolor{webgreen}{rgb}{0, 0.35, 0} % less intense green
\definecolor{webblue}{rgb}{0, 0, 0.50} % less intense blue
\definecolor{webred}{rgb}{0.35, 0, 0} % less intense red
%\usepackage[colorlinks=true,linkcolor=webblue,citecolor=webred,urlcolor=webgreen]{hyperref}
\usepackage{hyperref}
\hypersetup{
pdfmenubar=true,
pdftoolbar=true,
pdfpagemode={None}
colorlinks=true,
linkcolor=webblue,
citecolor=webblue,
urlcolor=webblue,
}
% \usepackage[pdfauthor={Boris Jeremic},
% colorlinks=true,
% linkcolor=webblue,
% citecolor=webblue,
% urlcolor=webblue,
% linktocpage,
% pdftex]{hyperref}
\usepackage{pause}
% or whatever
%\usepackage{html}
%\usepackage{url}
\usepackage[latin1]{inputenc}
% or whatever
\usepackage{times}
\usepackage[T1]{fontenc}
% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.
% Site Specific Dynamics of Structures:
%From Seismic Source to
%the Safety of Occupants and Content
\title[]
{Probabilistic Seismic Risk Analysis for\\
Inelastic Soil-Structure Systems }
%\subtitle
%{Include Only If Paper Has a Subtitle}
%\author[Author, Another] % (optional, use only with lots of authors)
%{F.~Author\inst{1} \and S.~Another\inst{2}}
% - Give the names in the same order as the appear in the paper.
% - Use the \inst{?} command only if the authors have different
% affiliation.
\pgfdeclareimage[height=0.2cm]{university-logo}{/home/jeremic/BG/amblemi/ucdavis_logo_blue_sm}
\pgfdeclareimage[height=0.7cm]{lbnl-logo}{/home/jeremic/BG/amblemi/lbnl-logo}
%\author[Jeremi{\'c} et al.] % (optional, use only with lots of authors)
\author[Jeremi{\'c} et al.] % (optional, use only with lots of authors)
%{Boris~Jeremi{\'c}}
{Boris Jeremi{\'c}
\\ \vspace*{1mm}
Hexiang Wang, Han Yang}
%\institute[Computational Geomechanics Group \hspace*{0.3truecm}
%\institute[\pgfuseimage{university-logo}\hspace*{0.1truecm}\pgfuseimage{lbnl-logo}] % (optional, but mostly needed)
\institute[\pgfuseimage{university-logo}] % (optional, but mostly needed)
%{ Professor, University of California, Davis\\
{ University of California, Davis}
% % 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 UCLA Lifelines Conference \\
Jan/Feb 2022}
\subject{}
% This is only inserted into the PDF information catalog. Can be left
% out.
% If you have a file called "university-logo-filename.xxx", where xxx
% is a graphic format that can be processed by latex or pdflatex,
% resp., then you can add a logo as follows:
%\pgfdeclareimage[height=0.2cm]{university-logo}{/home/jeremic/BG/amblemi/ucdavis_logo_gold_lrg}
%\logo{\pgfuseimage{university-logo}}
% \pgfdeclareimage[height=0.5cm]{university-logo}{university-logo-filename}
% \logo{\pgfuseimage{university-logo}}
% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
% \AtBeginSubsection[]
\setcounter{tocdepth}{3}
\AtBeginSubsection[]
% \AtBeginSection[]
{
\begin{scriptsize}
\begin{frame}
\frametitle{Outline}
\tableofcontents[currentsection,currentsubsection]
% \tableofcontents[currentsection]
\end{frame}
\end{scriptsize}
}
% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command:
\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\titlepage
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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{\ }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%dir
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Motivation}
\begin{itemize}
%\vspace*{0.3cm}
\item[] Improve modeling and simulation of infrastructure objects
% \vspace*{2mm}
% \item[] Expert numerical modeling and simulation tool
%
% \vspace*{1mm}
% \item[] Use of numerical models to
% analyze statics and dynamics of soil/rock-structure systems
%
\vspace*{4mm}
\item[] Modeling, epistemic uncertainty
\vspace*{4mm}
\item[] Parametric, aleatory uncertainty
\vspace*{4mm}
\item[] Goal is to Predict and Inform
% rather than (force) fit
%\vspace*{4mm}
% \item[] Engineer needs to know!
%
%
%
% \vspace*{1mm}
% \item[] Follow the flow, input and dissipation, of seismic energy,
% \vspace*{2mm}
% \item[]
% %System for
% {\bf Real}istic modeling and simulation of
% {\bf E}arthquakes and/or
% {\bf S}oils and/or
% {\bf S}tructures and their
% {\bf I}nteraction:\\
% Real-ESSI
% \hspace*{5mm}
% \url{http://real-essi.info/}
% % % % \hspace*{25mm}
% % \url{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}
% % % \href{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}{{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}
% % % % \url{http://ms-essi.info/}
% % %
%
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
%\frametitle{Parametric, Aleatory Uncertainty}
\frametitle{Aleatory Uncertainties, Material, Motions}
\vspace*{2mm}
%\vspace*{-5mm}
\begin{figure}[!hbpt]
\begin{center}
%
\hspace*{-7mm}
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01-Ed.pdf}
\hspace*{-3mm}
% \hfill
\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/YoungModulus_Histogram_Normal_01-Ed.pdf}
%
\end{center}
\end{figure}
\vspace*{-5mm}
%\vspace*{-1.8cm}
%\hspace*{-3.3cm}
\begin{flushright}
{\tiny
(cf. Phoon and Kulhawy (1999B))\\
~}
\end{flushright}
%
\vspace*{-9mm}
\begin{figure}[!hbpt]
\begin{center}
%
%\hspace*{-7mm}
\includegraphics[width=5.00truecm]{/home/jeremic/tex/works/Thesis/HexiangWang/time_series_motionsn_06ug2019_SMIRT/Acc_realization_200.pdf}
%\hspace*{-3mm}
%\includegraphics[width=2cm]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version2/Figures/Acc_time_series_realiztion70.pdf}
%\includegraphics[width=2cm]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version2/Figures/Acc_time_series_realiztion100.pdf}
%% \includegraphics[width=0.31\textwidth]{Figures/Acc_time_series_realiztion350.pdf}
%\includegraphics[width=2cm]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version2/Figures/Acc_time_series_realiztion367.pdf}
\includegraphics[width=4cm]{/home/jeremic/tex/works/Papers/2019/1D_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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Probabilistic Seismic Risk}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Stochastic Modeling}
\subsection{Uncertainty Propagation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Forward Uncertainty Propagation}
\begin{itemize}
\vspace*{3mm}
\item[-] Given uncertain material
\vspace*{3mm}
\item[-] Given uncertain loads
\vspace*{3mm}
\item[-] Determine uncertain response, $u_i, \dot{u}_i, \ddot{u}_i,
\epsilon_{ij}, \sigma_{ij}$, PDFs/CDFs
\vspace*{3mm}
\item[-] Direct, intrusive, analytic development
\vspace*{3mm}
\item[-] Circumvent Monte Carlo inefficiencies, inaccuracies
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Forward Uncertain Inelasticity}
%
% %
%
% \begin{itemize}
%
%
%
% \item[-] Incremental el--pl constitutive equation
% %
% \begin{eqnarray}
% \nonumber
% \Delta \sigma_{ij}
% =
% % E^{EP}_{ijkl}
% E^{EP}_{ijkl} \; \Delta \epsilon_{kl}
% =
% \left[
% E^{el}_{ijkl}
% -
% \frac{\displaystyle E^{el}_{ijmn} m_{mn} n_{pq} E^{el}_{pqkl}}
% {\displaystyle n_{rs} E^{el}_{rstu} m_{tu} - \xi_* h_*}
% \right]
% \Delta \epsilon_{kl}
% \end{eqnarray}
%
%
%
%
% \vspace*{2mm}
% \item[-] Dynamic Finite Elements
% %
% \begin{equation}
% { M} \ddot{ u_i} +
% { C} \dot{ u_i} +
% { K}^{ep} { u_i} =
% { F(t)}
% \nonumber
% \end{equation}
%
%
% \vspace*{2mm}
% \item[-] Material and loads are uncertain
%
%
%
% \end{itemize}
%
%
%
% \end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Probabilistic Elastic-Plastic Response}
%
%
% \begin{figure}[!hbpt]
% \begin{center}
% %\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDF-edited.pdf}
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2012/DOE-LLNL-workshop-27-28-Feb-2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_Contour_PDF-edited.pdf}
% \end{center}
% \end{figure}
%
% \end{frame}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{{Cam Clay with Random $G$, $M$ and $p_0$}}
\begin{figure}[!hbpt]
\begin{center}
\hspace*{-15mm}
\includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0-m.pdf}
%\hspace*{-2mm}
\includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourHighOCR_RandomG_RandomM-m.pdf}
\hspace*{-15mm}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}
% -- %%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Stochastic Elastic-Plastic FEM}
\vspace*{2mm}
Dynamic Finite Elements
$
{ M} \ddot{ u_i} +
{ C} \dot{ u_i} +
{ K}^{ep} { u_i} =
{ F(t)}$
\begin{itemize}
\vspace*{2mm}
\item[-] Input random field/process{\normalsize{(non-Gaussian, heterogeneous/ non-stationary)}}:
Multi-dimensional Hermite Polynomial Chaos (PC) with {known coefficients}
%\vspace{0.05in}
\vspace*{2mm}
\item[-] Output response process: Multi-dimensional Hermite PC with {unknown coefficients}
% \vspace{0.05in}
\vspace*{2mm}
\item[-] Galerkin projection: minimize the error to compute unknown coefficients of response process
% %\vspace{0.05in}
% \vspace*{2mm}
% \item[-] Time integration using Newmark's method
% % : Update coefficients following
% % an elastic-plastic constitutive law at each time step
\end{itemize}
%\scriptsize
%Note: PC = Polynomial Chaos
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
%\frametitle{State of the Art Probabilistic Seismic Risk Analysis}
\frametitle{Probabilistic Seismic Risk Analysis}
\begin{itemize}
% \item Target: safe design, acceptably small failure probability $\lambda (EDP)$
\item[-]
%Powerful tool
%Allows for
Objective, quantitative decision making based on exceedance rate
$\lambda (EDP>z)$
\item[-] PSRA: convolution of PSHA and fragility
% \[\lambda(EDP>z) = \int_{IM} \underbrace{|\frac{d\lambda(IM)}{dIM}|}_\text{PSHA} \underbrace{G(EDP|IM)}_{\text{fragility}} dIM \]
\vspace{-0.1cm}
\[\lambda(EDP>z) = \int \underbrace{|\frac{d\lambda(IM>x)}{dx}|}_\text{\textbf{PSHA}} \underbrace{G(EDP>z|IM=x)}_{\text{\textbf{fragility analysis}}} dx\]
\small{$\lambda(\cdot)$ : rate of exceedance\\
\vspace{0.07cm}
$EDP$: engineering demand parameter\\
\vspace{0.07cm}
$PSHA$: probabilistic seismic hazard analysis\\
\vspace{0.07cm}
$IM$: intensity measure, choice to be made (!) }
\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
% \item[-] IM choice sometimes contentious
%
% % \begin{itemize}
% % % \item[--] \small Geo-hazard: Liquefaction, slope deformation
% % % \item[--] \small PGA v.s. AI v.s. RMS for liquefaction
% % \item[--] \small AI v.s. PGV v.s. CAV for dam embankment
% % \end{itemize}
%
% \vspace{3mm}
%
% % \item Additional effort for new GMPEs
%
% % \begin{itemize}
% % \item[--] \small vector hazard: GMPE with covariance of IMs, fragility as function of IMs, rarely used
% % \end{itemize}
%
% % \item[-] Miscommunication: seismologists and engineers
% %
% % \begin{itemize}
% % \item[] \small $Sa(T_0)$ not compatible with time domain nonlinear analysis
% % \end{itemize}
%
% \end{itemize}
% \end{frame}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Risk Analysis Example}
% %\subsection{TDNIPSRA Formulation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Current State of Art Seismic Risk Analysis (SRA)}
%
%
% \begin{itemize}
% %\vspace{2mm}
%
% \item[-] Intensity measure (IM) selected as a proxy for ground motions,
% usually Spectral acceleration $Sa(T_0)$
%
% \vspace{4mm}
% \item[-] Ground Motion Prediction Equations (GMPEs) need development, ergodic or site specific
%
% \vspace{4mm}
% \item[-] Probabilistic seismic hazard analysis (PSHA)
% % for ground motion $\lambda(Sa>z)$
% % \begin{equation*}
% % \resizebox{0.85\hsize}{!}{%
% % $\lambda(Sa>z) = \sum_{i=1}^{NFL} \underbrace{N_i \int\int f_{mi}(M) f_{ri}(R|M)}_\text{seismic source characterization (SSC)} \underbrace{P(Sa>z|M, R)}_\text{GMPE} dM dR$}
% % \end{equation*}
%
% \vspace{4mm}
% \item[-] Fragility analysis $P(EDP>x|IM=z)$, deterministic time domain FEM,
% perhaps using Monte Carlo (MC)
%
% % \begin{itemize}
% %
% % \item[-] Records selection: Spectrum-matching technique UHS, etc
% %
% % \item[-] Incremental dynamic analysis: Monte Carlo
% %
% % \end{itemize}
%
%
%
% \end{itemize}
%
% % \begin{textblock}{15}(2.2, 9.2)
% % \begin{figure}[H]
% % \flushleft
% % % \includegraphics[width=0.38\linewidth]{pic/hazard_curve.png}
% % \includegraphics[width=0.38\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/hazard_curve.pdf}
% % \enspace
% % \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/design_spectra.png}
% % \end{figure}
% % \end{textblock}
%
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Issues in State-of-the-art SRA}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Seismic Risk Analysis Challenges}
%
%
% \begin{itemize}
%
%
%
% \item[-] IM serves as the proxy of damaging ground motions
% \vspace{2mm}
% \item[-] Does a single IM, e.g., $Sa(T_0)$, represent all uncertainty?
% %\item[-] Practically difficult/contentious to choose
%
%
% %\vspace{3mm}
% \vspace{2mm}
% \item[-] IMs difficult to choose, Spectral Acc, PGA, PGV...
%
%
% %%\vspace{3mm}
% %\item[-] Single IM does not contain all/most uncertainty
%
%
%
%
% \vspace{2mm}
% \item[-] Fragility analysis: incremental dynamic analysis (IDA)
% % using Monte Carlo method
%
% \vspace{2mm}
% \item[-] Use of Monte Carlo method, accuracy, efficiency...
%
% %\vspace{3mm}
% \vspace{2mm}
% \item[-] Monte Carlo, computationally expensive, CyberShake for LA, 20,000
% cases, 100Y runtime, (Maechling et al. 2007)
%
% %
% %
% % \vspace{3mm}
% % \item[-] Miscommunication between seismologists and struct/geotech engineers,
% % $Sa(T_0)$ not compatible with nonlinear FEM
%
%
%
%
% \end{itemize}
%
% \end{frame}
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Time Domain Intrusive PSRA Framework}
%
% %
%
% \begin{itemize}
%
% %\vspace*{2mm}
% \item[-] Stochastic Elastic-Plastic Finite Element Method, SEPFEM,
% ${M} \ddot{u_i} + {C} \dot{u_i} + {K}^{ep} {u_i} = {F(t)}$,
% (Sett et al. 2011)
%
%
% \vspace*{4mm}
% \item[-] Uncertain elastic-plastic material
% %stress and stiffness solution using
% %Forward Kolmogorov, Fokker-Planck equation
%
%
% \vspace*{4mm}
% \item[-] Uncertain seismic loads/motions
% % using Domain Reduction Method
%
%
% \vspace*{4mm}
% \item[-] Results, probability distribution functions for $\sigma_{ij}$,
% $\epsilon_{ij}$, $u_i$...
%
%
%
%
%
%
%
%
% \end{itemize}
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Stochastic Elastic-Plastic Finite Element Method}
%
%
%
% \begin{itemize}
%
% %\item[-] Material uncertainties: expanded along stochastic shape functions:
% \item[-] Material uncertainties: stochastic shape functions:
% $E^{ep}(x,t,\theta) = \sum_{i=0}^{P_d} E_i(x,t) * \Phi_i[\{\xi_1, ..., \xi_m\}]$
%
% \vspace*{1mm}
% \item[-] Loading uncertainties: stochastic shape functions
% $F(x,t,\theta) = \sum_{i=0}^{P_f} F_i(x,t) * \zeta_i[\{\xi_{m+1}, ..., \xi_f]$
%
% \vspace*{1mm}
% \item[-] Displacement expanded: stochastic shape functions:
% $u(x,t,\theta) = \sum_{i=0}^{P_u} u_i(x,t) * \Psi_i[\{\xi_1, ..., \xi_m, \xi_{m+1}, ..., \xi_f\}]$
%
%
% \vspace*{1mm}
% \item[-]
% Stochastic system of equations
% \vspace*{-2mm}
% \begin{tiny}
% \[
% \begin{bmatrix}
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\ \\
% \vdots & \vdots & \vdots & \vdots\\
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)} & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
% \end{bmatrix}
% \begin{bmatrix}
% u_{10} \\
% \vdots \\
% u_{N0}\\
% \vdots \\
% u_{1P_u}\\
% \vdots \\
% u_{NP_u}
% \end{bmatrix}
% =
% %\]
% %\[
% \begin{bmatrix}
% \sum_{i=0}^{P_f} f_i <\Psi_0\zeta_i> \\
% \sum_{i=0}^{P_f} f_i <\Psi_1\zeta_i> \\
% \sum_{i=0}^{P_f} f_i <\Psi_2\zeta_i> \\
% \vdots \\
% \sum_{i=0}^{P_f} f_i <\Psi_{P_u}\zeta_i>\\
% \end{bmatrix}
% \]
% \end{tiny}
%
%
%
%
%
% \end{itemize}
%
%
%
% \end{frame}
%
%
%
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% % \begin{frame}
% % \frametitle{Stochastic Elastic-Plastic Finite Element Method}
% % %\frametitle{SEPFEM : Formulation}
% %
% % Stochastic system of equations
% %
% % \begin{tiny}
% % \[
% % \begin{bmatrix}
% % \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
% % \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\ \\
% % \vdots & \vdots & \vdots & \vdots\\
% % \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)} & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
% % \end{bmatrix}
% % \begin{bmatrix}
% % u_{10} \\
% % \vdots \\
% % u_{N0}\\
% % \vdots \\
% % u_{1P_u}\\
% % \vdots \\
% % u_{NP_u}
% % \end{bmatrix}
% % =
% % %\]
% % %\[
% % \begin{bmatrix}
% % \sum_{i=0}^{P_f} f_i <\Psi_0\zeta_i> \\
% % \sum_{i=0}^{P_f} f_i <\Psi_1\zeta_i> \\
% % \sum_{i=0}^{P_f} f_i <\Psi_2\zeta_i> \\
% % \vdots \\
% % \sum_{i=0}^{P_f} f_i <\Psi_{P_u}\zeta_i>\\
% % \end{bmatrix}
% % \]
% % \end{tiny}
% %
% %
% %
% % % \normalsize{Typical number of terms required for a SEPFEM problem} \vspace{1cm}\\
% % \scalebox{0.7}{
% % \begin{tabular}{ c c c c}
% % \# KL terms material & \# KL terms load & PC order displacement& Total \# terms per DoF\\ \hline
% % 4 & 4 & 10 & 43758 \\
% % 4 & 4 & 20 & 3 108 105 \\
% % % 4 & 4 & 30 & 48 903 492 \\
% % 6 & 6 & 10 & 646 646 \\
% % % 6 & 6 & 20 & 225 792 840 \\
% % % 6 & 6 & 30 & 1.1058 $10^{10}$ \\
% % % 8 & 8 & 10 & 5 311 735 \\
% % % 8 & 8 & 20 & 7.3079 $10^{9}$ \\
% % % 8 & 8 & 30 & 9.9149 $10^{11}$\\
% %
% % ... & ... & ... & ...\\ \hline
% % \end{tabular}}
% %
% %
% % \end{frame}
% %
% %
% %
% %
%
%
%
%
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% %
% % \frametitle{Monte Carlo (MC)}
% %
% % \begin{itemize}
% %
% % \item Monte Carlo simulations: non-intrusive approach
% %
% % \begin{itemize}
% % \item [--] \small Slow convergence rate $1/\sqrt{N}$
% % \item [--] \small Hard for stable tail distribution toward low-risk level
% % \end{itemize}
% %
% % \item Fragility curve: incremental dynamic analysis (IDA)
% %
% % \begin{itemize}
% % \item [--] \small Impractical for large $3D$ nonlinear ESSI system
% % \end{itemize}
% %
% % \item Uncertain seismic wave propagation over regional geology
% %
% % \begin{itemize}
% % \item [--] \small CyberShake from SCEC
% % \item [--] \small Los Angeles, over 20,000 scenarios within 200 km, \textbf{300 million CPU-hours and over 100 years} (Maechling et al. 2007)
% % \end{itemize}
% %
% % \end{itemize}
% % \end{frame}
% %
% %
% %
% %
% %
% %
% %
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %\subsection{TDNIPSRA Example}
% \subsection[TDNIPSRA Example]{Example}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begingroup
\setbeamertemplate{footline}{}
\begin{frame}
%\frametitle{TDNIPSRA Framework}
\frametitle{Application: Seismic Hazard}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(0, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
\end{textblock}
\begin{textblock}{15}(0.3, 3.5)
\scriptsize{Seismic source characterization}
\end{textblock}
\begin{textblock}{15}(2.9, 5.2)
\tiny{UCERF3 (2014)}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(5.1, 6.5)
%$\Rightarrow$
{\Large $\rightarrow$}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(5.8, 3.9)
\vspace*{1mm}
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
\end{textblock}
\begin{textblock}{15}(7.1, 6.2)
\scalebox{.9}{\tiny{Fourier spectra}}
\\
\vspace*{-0.2cm}
\scalebox{.9}{\tiny{\hspace{0.14cm} Boore(2003)}}
\end{textblock}
\begin{textblock}{15}(6.1, 3.5)
\scriptsize{Stochastic ground motion}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(9.9, 6.5)
%{\bf $\Rightarrow$}
{\Large $\rightarrow$}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(10.5, 4.2)
% \includegraphics[width=0.35\linewidth]{pic/KL_exact_dis_correlation_from_dis.pdf}
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
\end{textblock}
\begin{textblock}{15}(11.1, 9.6)
\scriptsize{Uncertainty characterization \\
\hspace{0.1cm} Hermite polynomial chaos}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(10.2, 13.2)
{\Large $\leftarrow$}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(11, 11.2)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/structural_uncertainty.pdf}
\end{textblock}
\begin{textblock}{15}(5.3, 10.75)
\includegraphics[width=0.33\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/probabilsitc_evolution.png}
\end{textblock}
\begin{textblock}{15}(5.4, 9.6)
\scriptsize{\quad \quad Uncertainty propagation \\
\quad \quad \quad \quad SEPFEM}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(4.6, 13.2)
%$\Leftarrow$
{\Large $\leftarrow$}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(0.3, 11.0)
\includegraphics[width=0.29\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/seismic_risk_result_framework.png}
\end{textblock}
\begin{tikzpicture}[remember picture, overlay]
\draw[line width=1pt, draw=black, rounded corners=4pt, fill=gray!20, fill opacity=1]
([xshift=-25pt,yshift=-55pt]$(pic cs:a) + (0pt,8pt)$) rectangle ([xshift=95pt,yshift=-18pt]$(pic cs:b)+(0pt,-2pt)$);
\end{tikzpicture}
\begin{textblock}{15}(-0.1, 9.3)
\scriptsize
\quad \quad \quad \quad $\lambda(EDP>z)=$
$\quad \sum N_i(M_i, R_i) P(EDP>z|M_i, R_i)$
\end{textblock}
\begin{textblock}{15}(1.6, 10.7)
\scriptsize{EDP hazard/risk}
\end{textblock}
\end{frame}
\endgroup
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begingroup
%
% \setbeamertemplate{footline}{}
%
% \begin{frame}
%
% \frametitle{Time Domain Intrusive PSRA Framework}
%
%
% \begin{textblock}{15}(0, 4.0)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(0.3, 3.5)
% \scriptsize{Seismic source characterization}
% \end{textblock}
%
% \begin{textblock}{15}(2.9, 5.2)
% \tiny{UCERF3 (2014)}
% \end{textblock}
%
% \begin{textblock}{15}(5.1, 6.5)
% $\Rightarrow$
% \end{textblock}
%
%
% \begin{textblock}{15}(5.8, 3.9)
% \vspace*{1mm}
% \includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
% \end{textblock}
%
%
% \begin{textblock}{15}(7.1, 6.2)
% \scalebox{.9}{\tiny{Fourier spectra}}
% \\
% \vspace*{-0.2cm}
% \scalebox{.9}{\tiny{\hspace{0.14cm} Boore(2003)}}
% \end{textblock}
%
% \begin{textblock}{15}(6.1, 3.5)
% \scriptsize{Stochastic ground motion}
% \end{textblock}
%
% \begin{textblock}{15}(9.9, 6.5)
% $\Rightarrow$
% \end{textblock}
%
% \begin{textblock}{15}(10.5, 4.2)
% % \includegraphics[width=0.35\linewidth]{pic/KL_exact_dis_correlation_from_dis.pdf}
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(11.1, 9.6)
% \scriptsize{Uncertainty characterization \\
% \hspace{0.1cm} Hermite polynomial chaos}
% \end{textblock}
%
% \begin{textblock}{15}(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}(10.2, 13.2)
% % $\Leftarrow$
% % \end{textblock}
%
% % \begin{textblock}{15}(5.3, 10.75)
% % \includegraphics[width=0.33\linewidth]{pic/probabilsitc_evolution.png}
% % \end{textblock}
%
% % \begin{textblock}{15}(5.4, 9.6)
% % \scriptsize{\quad \quad Uncertainty propagation \\
% % \quad \quad \quad \quad stochastic FEM}
% % \end{textblock}
%
% % \begin{textblock}{15}(4.6, 13.2)
% % $\Leftarrow$
% % \end{textblock}
%
% % \begin{textblock}{15}(0.3, 11.0)
% % \includegraphics[width=0.29\linewidth]{pic/seismic_risk_result_framework.png}
% % \end{textblock}
%
% % \begin{tikzpicture}[remember picture, overlay]
% % \draw[line width=1pt, draw=black, rounded corners=4pt, fill=gray!20, fill opacity=1]
% % ([xshift=-25pt,yshift=-52pt]$(pic cs:a) + (0pt,8pt)$) rectangle ([xshift=95pt,yshift=-18pt]$(pic cs:b)+(0pt,-2pt)$);
% % \end{tikzpicture}
%
%
% % \begin{textblock}{15}(-0.1, 9.3)
% % \scriptsize
% % \quad \quad \quad \quad $\lambda(EDP>z)=$
%
% % $\quad \sum N_i(M_i, R_i) P(EDP>z|M_i, R_i)$
% % \end{textblock}
%
% % \begin{textblock}{15}(1.6, 10.7)
% % \scriptsize{EDP hazard/risk}
% % \end{textblock}
%
% \end{frame}
%
% \endgroup
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begingroup
%
% \setbeamertemplate{footline}{}
%
% \begin{frame}
%
% \frametitle{Time Domain Intrusive PSRA Framework}
%
%
% \begin{textblock}{15}(0, 4.0)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(0.3, 3.5)
% \scriptsize{Seismic source characterization}
% \end{textblock}
%
% \begin{textblock}{15}(2.9, 5.2)
% \tiny{UCERF3 (2014)}
% \end{textblock}
%
% \begin{textblock}{15}(5.1, 6.5)
% $\Rightarrow$
% \end{textblock}
%
%
% \begin{textblock}{15}(5.8, 3.9)
% \vspace*{1mm}
% \includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
% \end{textblock}
%
%
% \begin{textblock}{15}(7.1, 6.2)
% \scalebox{.9}{\tiny{Fourier spectra}}
% \\
% \vspace*{-0.2cm}
% \scalebox{.9}{\tiny{\hspace{0.14cm} Boore(2003)}}
% \end{textblock}
%
% \begin{textblock}{15}(6.1, 3.5)
% \scriptsize{Stochastic ground motion}
% \end{textblock}
%
% \begin{textblock}{15}(9.9, 6.5)
% $\Rightarrow$
% \end{textblock}
%
% \begin{textblock}{15}(10.5, 4.2)
% % \includegraphics[width=0.35\linewidth]{pic/KL_exact_dis_correlation_from_dis.pdf}
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(11.1, 9.6)
% \scriptsize{Uncertainty characterization \\
% \hspace{0.1cm} Hermite polynomial chaos}
% \end{textblock}
%
%
% \begin{textblock}{15}(10.2, 13.2)
% $\Leftarrow$
% \end{textblock}
%
% \begin{textblock}{15}(11, 11.2)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/structural_uncertainty.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(5.3, 10.75)
% \includegraphics[width=0.33\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/probabilsitc_evolution.png}
% \end{textblock}
%
% \begin{textblock}{15}(5.4, 9.6)
% \scriptsize{\quad \quad Uncertainty propagation \\
% \quad \quad \quad \quad stochastic FEM}
% \end{textblock}
%
% % \begin{textblock}{15}(4.6, 13.2)
% % $\Leftarrow$
% % \end{textblock}
%
% % \begin{textblock}{15}(0.3, 11.0)
% % \includegraphics[width=0.29\linewidth]{pic/seismic_risk_result_framework.png}
% % \end{textblock}
%
% % \begin{tikzpicture}[remember picture, overlay]
% % \draw[line width=1pt, draw=black, rounded corners=4pt, fill=gray!20, fill opacity=1]
% % ([xshift=-25pt,yshift=-52pt]$(pic cs:a) + (0pt,8pt)$) rectangle ([xshift=95pt,yshift=-18pt]$(pic cs:b)+(0pt,-2pt)$);
% % \end{tikzpicture}
%
%
% % \begin{textblock}{15}(-0.1, 9.3)
% % \scriptsize
% % \quad \quad \quad \quad $\lambda(EDP>z)=$
%
% % $\quad \sum N_i(M_i, R_i) P(EDP>z|M_i, R_i)$
% % \end{textblock}
%
% % \begin{textblock}{15}(1.6, 10.7)
% % \scriptsize{EDP hazard/risk}
% % \end{textblock}
%
% \end{frame}
%
% \endgroup
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begingroup
%
% \setbeamertemplate{footline}{}
%
% \begin{frame}
%
% \frametitle{Time Domain Intrusive PSRA Framework}
%
%
% \begin{textblock}{15}(0, 4.0)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(0.3, 3.5)
% \scriptsize{Seismic source characterization}
% \end{textblock}
%
% \begin{textblock}{15}(2.9, 5.2)
% \tiny{UCERF3 (2014)}
% \end{textblock}
%
% \begin{textblock}{15}(5.1, 6.5)
% $\Rightarrow$
% \end{textblock}
%
%
% \begin{textblock}{15}(5.8, 3.9)
% \vspace*{1mm}
% \includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
% \end{textblock}
%
%
% \begin{textblock}{15}(7.1, 6.2)
% \scalebox{.9}{\tiny{Fourier spectra}}
% \\
% \vspace*{-0.2cm}
% \scalebox{.9}{\tiny{\hspace{0.14cm} Boore(2003)}}
% \end{textblock}
%
% \begin{textblock}{15}(6.1, 3.5)
% \scriptsize{Stochastic ground motion}
% \end{textblock}
%
% \begin{textblock}{15}(9.9, 6.5)
% $\Rightarrow$
% \end{textblock}
%
% \begin{textblock}{15}(10.5, 4.2)
% % \includegraphics[width=0.35\linewidth]{pic/KL_exact_dis_correlation_from_dis.pdf}
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(11.1, 9.6)
% \scriptsize{Uncertainty characterization \\
% \hspace{0.1cm} Hermite polynomial chaos}
% \end{textblock}
%
%
% \begin{textblock}{15}(10.2, 13.2)
% $\Leftarrow$
% \end{textblock}
%
% \begin{textblock}{15}(11, 11.2)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/structural_uncertainty.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(5.3, 10.75)
% \includegraphics[width=0.33\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/probabilsitc_evolution.png}
% \end{textblock}
%
% \begin{textblock}{15}(5.4, 9.6)
% \scriptsize{\quad \quad Uncertainty propagation \\
% \quad \quad \quad \quad stochastic FEM}
% \end{textblock}
%
% \begin{textblock}{15}(4.6, 13.2)
% $\Leftarrow$
% \end{textblock}
%
% \begin{textblock}{15}(0.3, 11.0)
% \includegraphics[width=0.29\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/seismic_risk_result_framework.png}
% \end{textblock}
%
% \begin{tikzpicture}[remember picture, overlay]
% \draw[line width=1pt, draw=black, rounded corners=4pt, fill=gray!20, fill opacity=1]
% ([xshift=-25pt,yshift=-52pt]$(pic cs:a) + (0pt,8pt)$) rectangle ([xshift=95pt,yshift=-18pt]$(pic cs:b)+(0pt,-2pt)$);
% \end{tikzpicture}
%
%
% \begin{textblock}{15}(-0.1, 9.25)
% \scriptsize
% \quad \quad \quad \quad $\lambda(EDP>z)=$
%
% $\quad \sum N_i(M_i, R_i) P(EDP>z|M_i, R_i)$
% \end{textblock}
%
% \begin{textblock}{15}(1.6, 10.7)
% \scriptsize{EDP hazard/risk}
% \end{textblock}
%
% \end{frame}
%
% \endgroup
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% \begin{frame}
%
% \frametitle{Stochastic Ground Motion Modeling}
%
% % \vspace{-0.4cm}
%
% \begin{itemize}
%
% %\item[-] \normalsize Shift from modeling specific IM to fundamental characteristics of ground motions
% \item[-] \normalsize Shift from modeling specific IM to fundamental characteristics of ground motions
%
% \begin{itemize}
%
% \item[-] \normalsize Uncertain Fourier amplitude spectra (FAS)
%
% \item[-] \normalsize Uncertain Fourier phase spectra (FPS)
%
% \end{itemize}
%
% % \item \scriptsize Mean behavior of stochastic FAS
%
% % \begin{itemize}
%
% % \item[--] \scriptsize $w^2$ source radiation spectrum by \textit{Brune(1970)}
%
% % \item[--] \scriptsize Systematic studies by \textit{ \textbf{Boore}(1983, \textbf{2003}, 2015)}.
%
% % \end{itemize}
%
% \vspace{0.05cm}
%
% %\item[-] \normalsize Recent GMPE study of FAS,
% %(FAS marginal median \& variability GMPEs by \textit{{Bora et al.
% %(2018)}} and {\textit{Bayless \& Abrahamson (2019)}} ;
% %FAS Inter-frequency correlation GMPE by \textit{Stafford(2017)} and
% %{\textit{Bayless \& Abrahamson (2018)}})
%
%
% \vspace*{1mm}
% \item[-] \normalsize GMPE studies of FAS,
% (
% \textit{{Bora et al. (2018)}},
% \textit{Bayless \& Abrahamson (2018,2019)},
% \textit{Stafford(2017)},
% %{\textit{Bayless \& Abrahamson (2018)}
% )
%
% % \begin{itemize}
% %
% % \item[-] \scriptsize FAS marginal median \& variability GMPEs by \textit{\textbf{Bora et al. (2018)}} and \textbf{\textit{Bayless \& Abrahamson (2019)}}
% %
% % %\vspace{0.1cm}
% %
% % \item[-] \scriptsize FAS Inter-frequency correlation GMPE by \textit{Stafford(2017)} and \textbf{\textit{Bayless \& Abrahamson (2018)}}.
% %
% % \end{itemize}
%
%
% %\vspace{0.05cm}
%
% %\item[-] \normalsize Stochastic FPS by phase derivative (Boore,2005)
% %(Logistic phase derivative model by {\textit{Baglio \& Abrahamson (2017)}})
% \vspace*{1mm}
% \item[-] \normalsize Stochastic FPS by phase derivative (Boore,2005)
% (Logistic phase derivative model by {\textit{Baglio \& Abrahamson (2017)}})
%
% \vspace*{1mm}
% \item[-] \normalsize Near future change from \textbf{ $\boldsymbol{Sa(T_0)}$} to \textbf{FAS} and \textbf{FPS}
% %next five years
% % as envisioned by Abrahamson (2018)
%
% \end{itemize}
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %\subsection{Illustrative Example}
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
%\frametitle{TDNIPSRA Example Object}
\frametitle{Example Object}
\begin{textblock}{15}(0.5, 4.0)
\includegraphics[width=0.47\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/faults_configuration_new.pdf}
\end{textblock}
\begin{textblock}{15}(7.5, 3.4)
\includegraphics[width=0.55\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SSC_legend.pdf}
\end{textblock}
\begin{textblock}{15}(0.8, 11.6)
\scriptsize
\begin{itemize}
\item Fault 1: San Gregorio fault
\item Fault 2: Calaveras fault
\item Uncertainty: Segmentation, \\ slip rate, rupture geometry, etc.
\end{itemize}
\end{textblock}
\begin{textblock}{15}(8.5, 11.6)
\scriptsize
\begin{itemize}
\item 371 total seismic scenarios
\item $M \ 5 \sim 5.5$ and $6.5 \sim 7.0$
\item $R_{jb} \ 20km \sim 40km$
\end{itemize}
\end{textblock}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Stochastic Ground Motion Modeling}
\begin{textblock}{15}(1.0, 3.7)
\small Realizations of simulated uncertain motions for scenario $M=7$, $R=15km$:
\end{textblock}
\begin{textblock}{15}(0.5, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_time_series100.pdf}
\end{textblock}
\begin{textblock}{15}(5.5, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_time_series343.pdf} \enspace
\end{textblock}
\begin{textblock}{15}(10.5, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_time_series439.pdf}
\end{textblock}
\begin{textblock}{15}(1.0, 9.2)
\small Verification with GMPE:
\end{textblock}
\begin{textblock}{15}(0.3, 9.5)
\includegraphics[width=0.36\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SA_GMPE_verification_std_08_no_smooth.pdf}
\end{textblock}
\begin{textblock}{15}(5.6, 9.5)
\includegraphics[width=0.36\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Goodness_fit_std_08_no_smooth.pdf}
\end{textblock}
\begin{textblock}{15}(10.8, 9.5)
\includegraphics[width=0.36\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/Standard_deviation_std_08_no_smooth_new.pdf}
\end{textblock}
% \begin{textblock}{15}(0.5, 11.0)
% \begin{itemize}
% \item $\Delta \sigma= 84bar$, $\kappa=0.03s$ with total $\sigma=0.8ln$.
% \item Simulated median is not biased.
% \item Consistent total uncertainties with GMPE.
% \end{itemize}
% \end{textblock}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Stochastic Ground Motion Characterization}
{\begin{textblock}{15}(-0.1, 3.62)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_mean_acc_from_acc.pdf}
\quad \quad \quad Acc. marginal mean
\end{textblock}
\begin{textblock}{15}(3.7, 3.62)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_var_acc_from_acc.pdf}
\quad \quad \quad Acc. marginal S.D.
\end{textblock}
\begin{textblock}{15}(7.6, 3.8)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_exact_acc_correlation_from_acc.pdf}
\quad \quad \quad Acc. realization Cov.
\end{textblock}
\begin{textblock}{15}(11.8, 3.9)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_simulated_acc_correlation_from_acc.pdf}
\quad \quad Acc. synthesized Cov.
\end{textblock}}
\begin{textblock}{15}(-0.1, 9.3)
\scriptsize
\includegraphics[width=0.31\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_mean_dis_from_dis.pdf}
\end{textblock}
\begin{textblock}{15}(0.9, 13.75)
\scriptsize
Dis. marginal mean
\end{textblock}
\begin{textblock}{15}(4.2, 9.4)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_var_dis_from_dis.pdf}
\end{textblock}
\begin{textblock}{15}(5.1, 13.75)
\scriptsize
Dis. marginal S.D.
\end{textblock}
\begin{textblock}{15}(8.2, 9.5)
\scriptsize
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_exact_dis_correlation_from_dis.pdf}
\quad \quad Dis. realization Cov.
\end{textblock}
\begin{textblock}{15}(12.2, 9.6)
\scriptsize
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_simulated_dis_correlation_from_dis.pdf}
\quad Dis. synthesized Cov.
\end{textblock}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Stochastic Soil and Structure Modeling}
%Uncertain 1D shear response
\begin{figure}[!htbp]
\centering
%\subfloat[Uncertain $H_a$]{
%\hspace{-0.8cm}
%\includegraphics[width=0.53\textwidth]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/constitutive_relation_uncertainHa_certainCr_MC_verification.pdf}}
%\subfloat[Uncertain $H_a$ and $C_r$]{
%\hspace{-0.2cm}
%\includegraphics[width=0.53\textwidth]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/constitutive_relation_uncertainHa_uncertainCr_MC_verification.pdf}}
%\vspace{-2mm}
%\caption{\label{figure_probabilisitc_constitutive_relation} Intrusive probabilistic modeling of Armstrong-Frederick hysteretic behavior and verification with Monte Carlo simulation: (a) Gaussian distributed $Ha$ with mean 1.76 $\times 10^{7} \ N/m$ and 15\% coefficient of variation (COV), $C_r = 17.6$. (b) Gaussian distributed $Ha$ with mean 1.76 $\times 10^{7} \ N/m$ and 15\% coefficient of variation (COV), Gaussian distributed $C_r$ with mean 17.6 and 15\% COV.}
\subfloat[Frame]{
\hspace{-0.8cm}
\includegraphics[width=2.5cm]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/Shear-Frame-8-levels.jpg}}
\subfloat[Interstory response]{
\hspace{10mm}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Papers/2019/1D_risk/version6/Figures/constitutive_relation_uncertainHa_uncertainCr_MC_verification.pdf}}
%\vspace{-2mm}
%\caption{\label{figure_probabilisitc_constitutive_relation} Intrusive probabilistic modeling of Armstrong-Frederick hysteretic behavior and verification with Monte Carlo simulation: (a) Gaussian distributed $Ha$ with mean 1.76 $\times 10^{7} \ N/m$ and 15\% coefficient of variation (COV), $C_r = 17.6$. (b) Gaussian distributed $Ha$ with mean 1.76 $\times 10^{7} \ N/m$ and 15\% coefficient of variation (COV), Gaussian distributed $C_r$ with mean 17.6 and 15\% COV.}
\end{figure}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Probabilistic Dynamic Structural Response}
\begin{textblock}{15}(0.7, 4.5)
\scriptsize
\includegraphics[width=0.46\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/shear_frame_illustration_update.pdf}
\end{textblock}
\begin{textblock}{15}(0.2, 10.8)
\begin{itemize}
\scriptsize \item Coefficient of variation 15$\%$ for $H_a$ and $C_r$
%\scriptsize \item Exponential correlation with correlation \\
%length $l_c = 10$ floors
\scriptsize \item Time domain stochastic \\
El-Pl FEM analysis (SEPFEM)
% : uncertain \\ structure with uncertain excitations
\end{itemize}
\end{textblock}
\begin{textblock}{15}(7.7, 5)
\scriptsize
\includegraphics[width=0.52\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Probabilistic_Response_Node_1_new.pdf}
\end{textblock}
\begin{textblock}{15}(8.3, 4.2)
\scriptsize Probabilistic response of top floor from SFEM
\end{textblock}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begingroup
%
% \setbeamertemplate{footline}{}
%
% \begin{frame}
%
% \frametitle{Seismic Risk Analysis}
%
% \begin{textblock}{15}(1.9,3.8)
% \scriptsize
% \includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/MIDR_PDF_evolution.pdf}
% \includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/PDF_MIDR_combine.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(1.9,9.5)
% \scriptsize
% \includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/MIDR_distribution_different_floors.pdf}
% \includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/Risk_MIDR.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(0.8, 3.8)
% \scriptsize Engineering demand parameter (EDP): Maximum inter-story drift ratio (MIDR)
% \end{textblock}
% \end{frame}
% \endgroup
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begingroup
% \setbeamertemplate{footline}{}
% \begin{frame}
%
% \frametitle{Seismic Risk Analysis}
%
% \begin{textblock}{15}(1.9,3.8)
% \scriptsize
% \includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/PFA_distribution.pdf}
% \includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/Risk_PFA.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(1.9,9.3)
% \scriptsize
% \includegraphics[width=0.41\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/2D_EDP_PDF_1e5.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(8.7,9.4)
% \scriptsize
% \includegraphics[width=0.40\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/2D_EDP_PDF_downview_1e5.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(0.8, 3.8)
% \scriptsize Engineering demand parameter (EDP): Peak floor acceleration (PFA)
% \end{textblock}
% \end{frame}
% \endgroup
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Seismic Risk, Forward Analysis}
%\vspace{-0.5cm}
\vspace*{2mm}
\begin{itemize}
% \item[-] \small Damage measure (DM) defined on multiple EDPs:
% \item[-] \small Damage measure (DM) defined on single EDP:
\item[-] Damage measure defined on single EDP:
\vspace*{-3mm}
%\begin{textblock}{15}(0.7,8.9)
\begin{table}[!htbp]
\small
\resizebox{0.98\hsize}{!}{
\begin{tabular}{ccccccc}
%\hline
\textbf{DM} & MIDR\textgreater{}0.5\% & \textbf{MIDR\textgreater{}1\%} & MIDR\textgreater{}2\% & PFA\textgreater{}0.5${\rm m/s^2}$ & \textbf{PFA\textgreater{}1\boldsymbol{${\rm m/s^2}$}} & PFA\textgreater{}1.5${\rm m/s^2}$ \\
\hline
\textbf{Risk [/yr]} & 6.66$\times 10^{-3}$ & \textbf{3.83\boldsymbol{$\times 10^{-3}$}} & 9.97$\times 10^{-5}$ & 6.65$\times 10^{-3}$ & \textbf{1.92 \boldsymbol{$\times 10^{-3}$}} & 9.45$\times 10^{-5}$ \\
%\hline
\end{tabular}}
\end{table}
%\end{textblock}
\vspace{4mm}
\item[-] Damage measure (DM) defined on multiple EDPs:
% \vspace{2mm}
{\scriptsize $DM: \{\text{MIDR}>1\%\, \cup \,\text{PFA}>1{\rm m/s^2} \}$, seismic risk is \boldsymbol{$4.2 \times 10^{-3}/yr$} }
\vspace{1mm}
{\scriptsize $DM: \{\text{MIDR}>1\%\, \cap \,\text{PFA}>1{\rm m/s^2} \}$, seismic risk is \boldsymbol{$1.71 \times 10^{-3}/yr$}}
\vspace{3mm}
%\vspace{20mm}
\vspace{4mm}
\item[-] \small Seismic risk for DM defined on multiple EDPs can be quite
different from that defined on single EDP
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Forward Uncertain Inelasticity}
%
%
%
% \begin{itemize}
%
%
%
% \item[-] Incremental el--pl constitutive equation
% %
% \begin{eqnarray}
% \nonumber
% \Delta \sigma_{ij}
% =
% % E^{EP}_{ijkl}
% E^{EP}_{ijkl} \; \Delta \epsilon_{kl}
% =
% \left[
% E^{el}_{ijkl}
% -
% \frac{\displaystyle E^{el}_{ijmn} m_{mn} n_{pq} E^{el}_{pqkl}}
% {\displaystyle n_{rs} E^{el}_{rstu} m_{tu} - \xi_* h_*}
% \right]
% \Delta \epsilon_{kl}
% \end{eqnarray}
%
%
%
%
% \vspace*{2mm}
% \item[-] Dynamic Finite Elements
% %
% \begin{equation}
% { M} \ddot{ u_i} +
% { C} \dot{ u_i} +
% { K}^{ep} { u_i} =
% { F(t)}
% \nonumber
% \end{equation}
%
%
% \vspace*{2mm}
% \item[-] Material and loads are uncertain
%
%
%
% \end{itemize}
%
%
%
% \end{frame}
%
%
%
%
%
%
%
%
%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{{Cam Clay with Random $G$, $M$ and $p_0$}}
%
%
% \begin{figure}[!hbpt]
% \begin{center}
% \hspace*{-15mm}
% \includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0-m.pdf}
% %\hspace*{-2mm}
% \includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourHighOCR_RandomG_RandomM-m.pdf}
% \hspace*{-15mm}
% \end{center}
% \end{figure}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \end{frame}
% % -- %%%%%%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}{Time Domain Stochastic Galerkin Method}
%
%
% \vspace*{2mm}
% Dynamic Finite Elements
% $
% { M} \ddot{ u_i} +
% { C} \dot{ u_i} +
% { K}^{ep} { u_i} =
% { F(t)}$
%
%
% \begin{itemize}
% \vspace*{2mm}
% \item[-] Input random field/process{\normalsize{(non-Gaussian, heterogeneous/ non-stationary)}}:
% Multi-dimensional Hermite Polynomial Chaos (PC) with {known coefficients}
% %\vspace{0.05in}
% \vspace*{2mm}
% \item[-] Output response process: Multi-dimensional Hermite PC with {unknown coefficients}
% % \vspace{0.05in}
% \vspace*{2mm}
% \item[-] Galerkin projection: minimize the error to compute unknown coefficients of response process
% % %\vspace{0.05in}
% % \vspace*{2mm}
% % \item[-] Time integration using Newmark's method
% % % : Update coefficients following
% % % an elastic-plastic constitutive law at each time step
%
% \end{itemize}
%
% %\scriptsize
% %Note: PC = Polynomial Chaos
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% % \begin{frame}{Discretization of Input Random Process/Field $\beta(x,\theta)$}
% % \begin{center}
% % \includegraphics[scale=0.35]{/home/jeremic/tex/works/Thesis/FangboWang/slides_13Mar2019/Fangbo_slides/figs/PC_KL_explanation.PNG} \\
% % \end{center}
% %
% %
% % \footnotesize{Note: $\beta(x,\theta)$ is an input random process with any
% % marginal distribution, \\ \hspace{21mm} with any covariance structure;} \\
% % \footnotesize{\hspace{8mm} $\gamma(x,\theta)$ is a zero-mean unit-variance Gaussian random process.} \\
% %
% % \end{frame}
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}{Polynomial Chaos Representation}
%
% %\scriptsize{
% Material random field: \\
% %\vspace{-0.3cm}
% %\begin{equation*}
% $D(x, \theta)= \sum_{i=1}^{P1} a_i(x) \Psi_i(\left\{\xi_r(\theta)\right\})$
% %\end{equation*}
%
%
% \vspace{4mm}
%
% Seismic loads/motions random process: \\
% %\vspace{-0.3cm}
% %\begin{equation*}
% $f_m(t, \theta)=\sum_{j=1}^{P_2} f_{mj}(t) \Psi_j(\{\xi_k(\theta)\})$
% %\end{equation*}
%
% \vspace{4mm}
%
% Displacement response: \\
% %\vspace{-0.3cm}
% %\begin{equation*}
% $u_n(t, \theta)=\sum_{k=1}^{P_3} d_{nk}(t) \Psi_k(\{\xi_l(\theta)\})$
% %\end{equation*}
%
% \vspace{3mm}
%
% %Acceleration response:
% %%\vspace{-0.3cm}
% %%\begin{equation*}
% %$\ddot u_n(t, \theta)=\sum_{k=1}^{P_3} \ddot d_{nk}(t) \Psi_k(\{\xi_l(\theta)\})$
% %%\end{equation*}
%
% %\vspace{3mm}
% \vspace{5mm}
%
% where $a_i(x), f_{mj}(t)$ are {known PC coefficients}, while $d_{nk}(t)$
% are {unknown PC coefficients}.
% %}
%
% \end{frame}
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \subsection{Direct Solution for Probabilistic Stiffness and Stress in 1D}
% %
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%% BEGGINING PEP %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% \begin{frame}{Direct Probabilistic Constitutive Solution in 1D}
%
%
% % \begin{itemize}
% %
% % \vspace{0.5cm}
% %
% % \item<1-> Probabilistic constitutive modeling : \vspace{0.5cm}
%
% \begin{itemize}
%
%
% \vspace*{4mm}
% \item[-] Zero elastic region elasto-plasticity with stochastic Armstrong-Frederick
% kinematic hardening
%
% $ \Delta\sigma =\ H_a \Delta \epsilon - c_r \sigma |\Delta \epsilon| ;
% \hspace{0.5cm}
% E_t = {d\sigma}/{d\epsilon} = H_a \pm c_r \sigma $
%
% \vspace*{4mm}
% \item[-] Uncertain:
% init. stiff. $H_a$,
% shear strength $H_a/c_r$,
% strain $\Delta \epsilon$:
%
% $ H_a = \Sigma h_i \Phi_i; \;\;\;
% C_r = \Sigma c_i \Phi_i; \;\;\;
% \Delta\epsilon = \Sigma \Delta\epsilon_i \Phi_i $
%
%
%
% \vspace*{4mm}
% \item[-] Resulting stress and stiffness are also uncertain
%
% % -
% % - $ \sum_{l=1}^{P_{\sigma}} \Delta\sigma_i \Phi_i = \sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k \Phi_i \Phi_k - \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_i \Delta \epsilon_k \sigma_l \Phi_j \Phi_k \Phi_l$
% % -
% % - $ \sum_{l=1}^{P_{E_t}} \Delta E_{t_i} \Phi_i = \sum_{i=1}^{P_h} h_i \Phi_i \pm \sum_{i=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_i \sigma_l \Phi_i \Phi_l$
% % -
%
%
% \end{itemize}
%
%
% % \vspace{0.5cm}
%
%
%
% % \vspace{1cm}
%
% %\item<1-> Time integration is done via Newmark algorithm
%
% %
% % \end{itemize}
% %
% \end{frame}
%
%
% % % % % % % % % % % % % % % % %
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}{Direct Probabilistic Stiffness Solution}
%
% \begin{itemize}
%
%
% \item[-] Analytic product for all the components,
%
% $ E^{EP}_{ijkl}
% =
% \left[
% E^{el}_{ijkl}
% -
% \frac{\displaystyle E^{el}_{ijmn} m_{mn} n_{pq} E^{el}_{pqkl}}
% {\displaystyle n_{rs} E^{el}_{rstu} m_{tu} - \xi_* h_*}
% \right]
% $
%
%
%
%
% \vspace*{2mm}
% \item[-] Stiffness: each Polynomial Chaos component is updated incrementally
% % at each Gauss Point via stochastic Galerkin projection
%
%
%
% \small{$E_{t_1}^{n+1} = \frac{1}{<\Phi_1\Phi_1> }\{\sum_{i=1}^{P_h} \ h_i <\Phi_i \Phi_1> \pm \sum_{j=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_j \sigma_l^{n+1} <\Phi_j \Phi_l \Phi_1>\}$}
% \\
% . . .
% %
% %
% % $\large{\vdots}$
% \\
% \small{$E_{t_P}^{n+1} = \frac{1}{<\Phi_1\Phi_P> }\{\sum_{i=1}^{P_h} \ h_i <\Phi_i \Phi_P> \pm \sum_{j=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_j \sigma_l^{n+1} <\Phi_j \Phi_l \Phi_P>\}$}
%
%
% \vspace*{2mm}
% \item[-] Total stiffness is :
%
% $ E_{t}^{n+1} = \sum_{l=1}^{P_{E}} E_{t_i}^{n+1} \Phi_i $
%
%
%
%
% \end{itemize}
%
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}{Direct Probabilistic Stress Solution}
%
% \begin{itemize}
%
%
%
% \item[-] Analytic product, for each stress component,
%
% $ \Delta \sigma_{ij} = E^{EP}_{ijkl} \; \Delta \epsilon_{kl} $
% % =
% % \left[
% % E^{el}_{ijkl}
% % -
% % \frac{\displaystyle E^{el}_{ijmn} m_{mn} n_{pq} E^{el}_{pqkl}}
% % {\displaystyle n_{rs} E^{el}_{rstu} m_{tu} - \xi_* h_*}
% % \right]
% % \Delta \epsilon_{kl}
% %
%
%
% \vspace*{2mm}
% \item[-] Incremental stress: each Polynomial Chaos component is updated
% incrementally
% % via stochastic Galerkin projection
%
%
%
%
% {$\Delta\sigma_1^{n+1} = \frac{1}{<\Phi_1\Phi_1> }\{\sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k^n <\Phi_i \Phi_k \Phi_1>- \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_j \Delta \epsilon_k^n \sigma_l^n <\Phi_j \Phi_k \Phi_l \Phi_1>\}$}
% \\
% . . .
% \\
% % ${\vdots}$
% {$\Delta\sigma_P^{n+1} = \frac{1}{<\Phi_P\Phi_P> }\{\sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k^n <\Phi_i \Phi_k \Phi_P>- \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_j \Delta \epsilon_k^n \sigma_l^n <\Phi_j \Phi_k \Phi_l \Phi_P>\}$}
%
%
% \vspace*{2mm}
% \item[-] Stress update:
%
% $ \sum_{l=1}^{P_{\sigma}} \sigma_i^{n+1} \Phi_i = \sum_{l=1}^{P_{\sigma}} \sigma_i^{n} \Phi_i + \sum_{l=1}^{P_{\sigma}} \Delta\sigma_i^{n+1} \Phi_i$
%
%
%
% \end{itemize}
%
% \end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
% \frametitle{Probabilistic Elastic-Plastic Response}
%
%
% % % \vspace*{-5mm}
% % \begin{center}
% % % \hspace*{-15mm}
% % \movie[label=show3,width=7cm,poster,autostart,showcontrols]
% % {\includegraphics[width=7cm]
% % {/home/jeremic/tex/works/Thesis/HanYang/Files_06June2017/DOE_Annual_2017/Figures/NPP_Plastic_Dissipation_Density.png}}
% % %{/home/jeremic/tex/works/Thesis/HanYang/Files_06June2017/DOE_Annual_2017/Figures/NPP_without_Contact_vonMises.mp4}
% % {NPP_without_Contact_vonMises.mp4}
% % \end{center}
%
% %\vspace*{-5mm}
% \begin{center}
% % \hspace*{-15mm}
% \movie[label=show3,width=9cm,poster,autostart,showcontrols]
% {\includegraphics[width=9cm]
% {/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.png}}
% % /home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.pdf
% %{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Animations/PEP_Animation.mp4}
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/PEP_Animation.mp4}
% \end{center}
%
%
%
%
% \begin{flushleft}
% \vspace*{-15mm}
% \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/PEP_Animation.mp4}
% % \href{./homo_50m-mesh_45degree_Ormsby.mp4}
% {\tiny (MP4)}
% \end{flushleft}
% %
%
%
%
%
% %
% % \includegraphics[width = 12cm]{./img/figure_PEP_25.pdf}
%
%
% \end{frame}
%
%
%
%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %\section[Formulation]{Stochastic Dynamic Finite Element Formulation}
% %\subsection[Time domain stochastic Galerkin method]{Time domain stochastic Galerkin method}
% %\frame{\tableofcontents[currentsubsection,sectionstyle=show/shaded]}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Stochastic Elastic-Plastic Finite Element Method}
%
%
%
% \begin{itemize}
%
% \item[-] Material uncertainty expanded into stochastic shape funcs.
% %$E(x,t,\theta) = \sum_{i=0}^{P_d} r_i(x,t) * \Phi_i[\{\xi_1, ..., \xi_m\}]$
%
% \vspace*{1mm}
% \item[-] Loading uncertainty expanded into stochastic shape funcs.
% %$f(x,t,\theta) = \sum_{i=0}^{P_f} f_i(x,t) * \zeta_i[\{\xi_{m+1}, ..., \xi_f]$
%
% \vspace*{1mm}
% \item[-] Displacement expanded into stochastic shape funcs.
% %$u(x,t,\theta) = \sum_{i=0}^{P_u} u_i(x,t) * \Psi_i[\{\xi_1, ..., \xi_m, \xi_{m+1}, ..., \xi_f\}]$
%
% %\item
% %Stochastic system of equation resulting from Galerkin approach (static example):
% %
% %\item Time domain integration using Newmark and/or HHT, in probabilistic spaces
%
%
%
% \vspace*{1mm}
% \item[-] Jeremi{\'c} et al. 2011
%
%
% \end{itemize}
%
%
% \begin{tiny}
% \[
% %$
% \begin{bmatrix}
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\ \\
% \vdots & \vdots & \vdots & \vdots\\
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)} & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
% \end{bmatrix}
% \begin{bmatrix}
% \Delta u_{10} \\
% \vdots \\
% \Delta u_{N0}\\
% \vdots \\
% \Delta u_{1P_u}\\
% \vdots \\
% \Delta u_{NP_u}
% \end{bmatrix}
% =
% %\]
% %\[
% \begin{bmatrix}
% \sum_{i=0}^{P_f} f_i <\Psi_0\zeta_i> \\
% \sum_{i=0}^{P_f} f_i <\Psi_1\zeta_i> \\
% \sum_{i=0}^{P_f} f_i <\Psi_2\zeta_i> \\
% \vdots \\
% \sum_{i=0}^{P_f} f_i <\Psi_{P_u}\zeta_i>\\
% \end{bmatrix}
% %$
% \]
% \end{tiny}
%
%
%
%
%
% \end{frame}
%
%
%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
% \frametitle{SEPFEM: System Size}
%
% \begin{itemize}
%
% \item[-] SEPFEM offers a complete probabilistic solution
%
% \item[-] It is NOT based on Monte Carlo approach
%
% \item[-] System of equations grows (!)
%
% \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}
%
%
%
%
%
%
%
%
%
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% \subsection{Backward Uncertainty Propagation, Sensitivities}
% %\subsection{Sobol Sensitivity Analysis}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Backward Uncertainty Propagation, Sensitivities}
%
% \begin{itemize}
%
% \vspace*{2mm}
% \item[-] Given forward uncertain response, PDFs, CDFs...
%
% \vspace*{6mm}
% \item[-] Contributions of uncertain input to forward uncertainties
%
% \vspace*{6mm}
% \item[-] Sensitivity of forward uncertain response to input uncertainties
%
%
% \end{itemize}
%
%
% \end{frame}
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{ANOVA Representation}
%
%
%
%
%
%
% %\vspace*{10mm}
%
% Model with $n$ uncertain inputs ($\boldsymbol{x}$) and scalar output $y$:
%
%
% \vspace*{-6mm}
% \begin{equation}
% y = f(\boldsymbol{x}) \mbox{;} \ \ \boldsymbol{x} \in I^{n}
% \nonumber
% \end{equation}
%
% % input parameters $\boldsymbol{x}$ are defined in $n$ dimensional unit
% % cube $I^{n}$
% %
%
% \vspace*{5mm}
%
% The ANalysis Of VAriance representation
% % of $f(x)$
% (Sobol 2001):
%
% \vspace*{-6mm}
% \begin{eqnarray*}
% f(x_1, ... x_n) = f_0 + \sum_{i=1}^{n} f_i(x_i) +
% \sum_{1\leq i