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\usepackage{epstopdf}
% %% Jose Antonio Abell Mena provided this for DSL descriptions
% % (used in a file _Chapter_SoftwareHardware_Domain_Specific_Language_English.tex
% % This is added for listing FEI DSL
% % since he customized it, it needs to be changed (linked to
% % /usr/share/texmf/tex/latex/misc)
% %\usepackage{myListings}
% \input{essi_listings_options.tex}
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% for tikzpicture
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%\usepackage{MnSymbol}%
% for smileys
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% for subfloat figures:
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%used for truncated section title in headers
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\usepackage{amsmath}
\usepackage{amsmath}
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\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}
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% for listing DSL
\input{essi_listings_options.tex}
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% for inclusion of other PDF pages, in this case Frank's presentation
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%This is a macro to convert eps to pdf files on the fly.
% make sure figure syntax uses graphicx syntax NOT epsfig syntax
%from http://mailman.mit.edu/pipermail/macpartners/2005-January/000780.html
%
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% % we are running pdflatex, so convert .pdf files to .pdf
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% %*****************************************
%% ovo je za cirilicu
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% \usetheme{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
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% \usetheme{Montpellier} % ima graf sadrzaj gore
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%%%% HYPERREF HYPERREF HYPERREF HYPERREF HYPERREF
%%%% HYPERREF HYPERREF HYPERREF HYPERREF HYPERREF
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% \usepackage[pdfauthor={Boris Jeremic},
% colorlinks=true,
% linkcolor=webblue,
% citecolor=webblue,
% urlcolor=webblue,
% linktocpage,
% pdftex]{hyperref}
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% or whatever
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% or whatever
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% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.
% Site Specific Dynamics of Structures:
%From Seismic Source to
%the Safety of Occupants and Content
\title[Engineering Analysis Toolbox]
{Engineering Analysis Toolbox
\\
The Real-ESSI Simulator System}
%\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)
\author[Jeremi{\'c} et al.] % (optional, use only with lots of authors)
%{Boris~Jeremi{\'c}}
{Boris Jeremi{\'c}
~~~~
{\cyss Boris Jeremi\cj{}}
}
%\\
%{\cyss Boris Jeremi\cj{}}
% }
% \\
%Han Yang, Hexiang Wang, Sumeet Kumar Sinha }
%\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, CA, USA}
% % 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 $5^{th}$ SEECCM
{ $5^{th}$ SEECCM
\\
July 2023
\\ ~ \\
{\cyss Vrnjachka Banja, Srbija}}
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\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}{1}
% \AtBeginSubsection[]
\AtBeginSection[]
{
\begin{scriptsize}
\begin{frame}
\frametitle{Outline}
% \tableofcontents[currentsection,currentsubsection]
\tableofcontents[currentsection]
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}
% % 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]
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% }
% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command:
\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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %\subsection{Motivation}
% \subsection{\ }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%dir
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Motivation}
\begin{itemize}
\vspace*{1mm}
\item[-] Safety and economy of infrastructure
%\vspace*{0.3cm}
%\vspace*{2mm}
% \item[] Improve analysis of infrastructure objects
% %\vspace*{1mm}
% % \item[-] Analysis of Earthquakes and/or Soils and/or Structures and their Interaction
% % \item[-] Analysis of Earthquakes, Soils, Structures and their Interaction
% % \item[-] Soils, Structures and their Interaction, statics and dynamics
% \item[-] Soils, Structures, statics and dynamics
% \vspace*{1mm}
% \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*{1mm}
% \item[-] Analysis to predict and inform
% % rather than (force) fit
%\vspace*{1mm}
% \item[-] Design, build and maintain sustainable objects
\vspace*{4mm}
\item[-] Design, build and maintain sustainable infrastructure
\vspace*{4mm}
\item[-] Responsible Engineer, with Executive Powers
%\vspace*{1mm}
% \item[] Engineer with executive powers
%\vspace*{2mm}
\vspace*{4mm}
\item[-] Engineer with versatile, quality assured analysis tool to
\begin{itemize}
\vspace*{1mm}
% \item[-] Explore various design concepts
\item[-] Explore design concepts
\vspace*{1mm}
\item[-] Assess infrastructure performance
\end{itemize}
% \vspace*{2mm}
% \item[-] Choice of analysis/modeling level of sophistication
%
% \vspace*{1mm}
% \item[] Modeling simplifications, Epistemic uncertainty
%
% \vspace*{1mm}
% \item[] Variable, random material and loads, Aleatory uncertainty
%
\vspace*{4mm}
\item[-] Engineering Analysis to Predict and Inform
%
%
%
% \vspace*{1mm}
% \begin{figure}[!hbpt]
% \begin{center}
% %
% %\hspace*{-7mm}
% %\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Conferences/2021/CU-Boulder-GEGM-seminar-series-02Apr2021/present/Saint_Sophia_Constantinopolis.jpg}
% %\hspace*{1mm}
% \includegraphics[height=1.2truecm]{/home/jeremic/tex/works/Conferences/2023/5_SEECCM_Vrnjacka_Banja_5-6Jul2023/present/Parthenon_on_Acropolis_June2023.jpg}
% %\hspace*{2mm}
% \hspace*{2mm}
% \includegraphics[height=1.2truecm]{/home/jeremic/tex/works/Conferences/2021/ASCE-4_Kennedy_Lecture/present/Aya-Sofia_03_1990.jpg}
% %\hspace*{2mm}
% \hspace*{2mm}
% \includegraphics[height=1.2truecm]{/home/jeremic/tex/works/Conferences/2021/ASCE-4_Kennedy_Lecture/present/ZhaozhouBridge.jpg}
% \hspace*{2mm}
% %\hspace*{2mm}
% \includegraphics[height=1.2truecm]{/home/jeremic/tex/works/Conferences/2021/ASCE-4_Kennedy_Lecture/present/Four_Water_Wheels_Hama_Syria.jpg}
% \hspace*{2mm}
% %\hspace*{2mm}
% \includegraphics[height=1.2truecm]{/home/jeremic/tex/works/Conferences/2023/5_SEECCM_Vrnjacka_Banja_5-6Jul2023/present/Eiffel_tower_design_drawings/Eiffel_tower_02.jpg}
% %
% %\vspace*{3mm}
% \end{center}
% \end{figure}
% %\vspace*{3mm}
%
% %\vspace*{1mm}
% \begin{figure}[!hbpt]
% \begin{center}
% %
% %\hspace*{-7mm}
% %\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Conferences/2021/CU-Boulder-GEGM-seminar-series-02Apr2021/present/Saint_Sophia_Constantinopolis.jpg}
% %\includegraphics[height=1.8truecm]{/home/jeremic/tex/works/Conferences/2021/ASCE-4_Kennedy_Lecture/present/Aya-Sofia_03_1990.jpg}
% %\hspace*{2mm}
% \includegraphics[height=3.5truecm]{/home/jeremic/tex/works/Conferences/2023/5_SEECCM_Vrnjacka_Banja_5-6Jul2023/present/Eiffel_tower_design_drawings/Eiffel_tower_force_diagram_shape.jpg}
% \hspace*{4mm}
% \includegraphics[height=3.5truecm]{/home/jeremic/tex/works/Conferences/2023/5_SEECCM_Vrnjacka_Banja_5-6Jul2023/present/Eiffel_tower_design_drawings/Eiffel_tower_foundation.jpg}
% %\hspace*{2mm}
% %
% %\vspace*{3mm}
% \vspace*{-2mm}
% \end{center}
% \end{figure}
% %\vspace*{3mm}
%
%
%
% \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.us/}
% % % % \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.us/}
% % %
%
%
%
%
% \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{Engineer Needs to Know!}
% \begin{itemize}
%
%
%
%
% \vspace*{2mm}
% \item[-] Responsible Engineer, with executive powers
%
% %\vspace*{1mm}
% % \item[] Engineer with executive powers
%
%
% %\vspace*{2mm}
% \vspace*{4mm}
% \item[-] Engineer with versatile, quality assured analysis tool to
%
% \begin{itemize}
% \vspace*{1mm}
% % \item[-] Explore various design concepts
% \item[-] Explore design concepts
%
% \vspace*{1mm}
% \item[-] Assess infrastructure performance
%
% \end{itemize}
%
%
% \vspace*{4mm}
% \item[-] Choice of analysis/modeling level of sophistication
% %
% % \vspace*{1mm}
% % \item[] Modeling simplifications, Epistemic uncertainty
% %
% % \vspace*{1mm}
% % \item[] Variable, random material and loads, Aleatory uncertainty
% %
%
%
%
% \vspace*{4mm}
% \item[-] Analysis to Predict and Inform
%
%
%
%
%
%
% \end{itemize}
%
\vspace*{1mm}
\begin{figure}[!hbpt]
\begin{center}
%
%\hspace*{-7mm}
%\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Conferences/2021/CU-Boulder-GEGM-seminar-series-02Apr2021/present/Saint_Sophia_Constantinopolis.jpg}
%\hspace*{1mm}
%\includegraphics[height=2.0truecm]{/home/jeremic/tex/works/Conferences/2023/5_SEECCM_Vrnjacka_Banja_5-6Jul2023/present/Parthenon_on_Acropolis_June2023.jpg}
%\hspace*{2mm}
%\hspace*{2mm}
\includegraphics[height=2.0truecm]{/home/jeremic/tex/works/Conferences/2021/ASCE-4_Kennedy_Lecture/present/Aya-Sofia_03_1990.jpg}
%\hspace*{2mm}
\hspace*{2mm}
\includegraphics[height=2.0truecm]{/home/jeremic/tex/works/Conferences/2021/ASCE-4_Kennedy_Lecture/present/ZhaozhouBridge.jpg}
%\hspace*{2mm}
\includegraphics[height=2.0truecm]{/home/jeremic/tex/works/Conferences/2021/ASCE-4_Kennedy_Lecture/present/Four_Water_Wheels_Hama_Syria.jpg}
%\hspace*{2mm}
%%\hspace*{2mm}
%\includegraphics[height=1.8truecm]{/home/jeremic/tex/works/Conferences/2023/5_SEECCM_Vrnjacka_Banja_5-6Jul2023/present/Eiffel_tower_design_drawings/Eiffel_tower_02.jpg}
%
%\vspace*{3mm}
\end{center}
\end{figure}
%\vspace*{3mm}
%\vspace*{1mm}
\begin{figure}[!hbpt]
\begin{center}
%
%\hspace*{-7mm}
%\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Conferences/2021/CU-Boulder-GEGM-seminar-series-02Apr2021/present/Saint_Sophia_Constantinopolis.jpg}
%\includegraphics[height=1.8truecm]{/home/jeremic/tex/works/Conferences/2021/ASCE-4_Kennedy_Lecture/present/Aya-Sofia_03_1990.jpg}
%\hspace*{2mm}
\includegraphics[height=3.5truecm]{/home/jeremic/tex/works/Conferences/2023/5_SEECCM_Vrnjacka_Banja_5-6Jul2023/present/Eiffel_tower_design_drawings/Eiffel_tower_force_diagram_shape.jpg}
\hspace*{2mm}
\includegraphics[height=3.5truecm]{/home/jeremic/tex/works/Conferences/2023/5_SEECCM_Vrnjacka_Banja_5-6Jul2023/present/Eiffel_tower_design_drawings/Eiffel_tower_02.jpg}
\hspace*{2mm}
\includegraphics[height=3.5truecm]{/home/jeremic/tex/works/Conferences/2023/5_SEECCM_Vrnjacka_Banja_5-6Jul2023/present/Eiffel_tower_design_drawings/Eiffel_tower_foundation.jpg}
%\hspace*{2mm}
%
%\vspace*{3mm}
\vspace*{-2mm}
\end{center}
\end{figure}
%\vspace*{3mm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
%\frametitle{ESSI Challenges}
\frametitle{Civil Engineering Analysis Challenges}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{figure}[!htb]
\begin{center}
% %\includegraphics[width=5cm]{Figure-files/_Chapter_Applications_ESSI_BOOK/Building_modeling_Issues_01_pdf.pdf}
%\vspace*{-3mm}
\includegraphics[width=2.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Seismic_Motions_modeling_Issues_01_pdf.pdf}
\includegraphics[width=5.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Seismic_Motions_modeling_Issues_02_pdf.pdf}
\\
%\vspace*{-2mm}
%\vspace*{-8mm}
\vspace*{2mm}
\includegraphics[width=3cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/NPP_Modeling_Issues_03.jpg}
\includegraphics[width=2.2cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/SMR_Modeling_Issues_02_pdf.pdf}
\includegraphics[width=2.8cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Building_modeling_Issues_03_pdf.pdf}
\includegraphics[width=2.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Building_modeling_Issues_02_pdf.pdf}
\\
\vspace*{2mm}
\includegraphics[width=2.5cm]{/home/jeremic/tex/Classes/2020/Spring_semester_ETH/ESSI/Term_Projects_from_Students/Tunnel02.pdf}
\includegraphics[width=1.2cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Dam_modeling_Issues_01_pdf.pdf}
\includegraphics[width=3.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Dam_modeling_Issues_02_pdf.pdf}
\includegraphics[width=2.0cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Bridge_modeling_Issues_01_pdf.pdf}
% %\vspace*{-8mm}
% \caption{\label{ESSI_Models_and_Challenges}
% ESSI modeling and simulation challenges:
% Free field motions, 3C/6C vs 3$\times$1C;
% Nuclear Power Plant structure -- soil/rock system, Small Modular Reactor structure -- soil/rock system;
% Low and High Building-foundation-soil system;
% Dam-Foundation-Fluid system;
% Bridge-soil system;
% %
% Aspects of modeling:
% 1) Seismic motions,
% 2) Inelastic soil and rock,
% 3) Inelastic interface/contact/joints, foundation with soil/rock and
% interfaces/contacts/joints within structure,
% 4) Inelastic structure, systems and components,
% 5) Solid, Structure -- Fluid interaction, external (reservoirs, fluid pools...) and internal
% (fully saturated and partially, (un-)saturated soil, rock and concrete).}
\end{center}
\end{figure}
%
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% \begin{itemize}
%
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% %\vspace*{1mm}
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% (cf. Phoon and Kulhawy (1999B))\\
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\section{Engineering Analysis Methods and Tools}
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% \subsection{Real-ESSI }
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\begin{frame}
\frametitle{Engineering Analysis System}
%\frametitle{Real-ESSI Simulator Analysis System}
\begin{itemize}
\vspace*{1mm}
\item[-]
Statics and dynamics of rock, soil, structures, fluids...
\vspace*{1mm}
\item[-]
Linear, Nonlinear, Inelastic
\vspace*{1mm}
\item[-]
Deterministic and Probabilistic
\vspace*{1mm}
\item[-]
High Performance Computing, HPC
\vspace*{1mm}
\item[-]
% Reduction of modeling, epistemic uncertainty
Reduction of Modeling Uncertainty
\vspace*{1mm}
\item[-]
% Propagation of parametric, aleatory uncertainty
Propagation of Parametric Uncertainty
\vspace*{1mm}
\item[-]
% Quality assurance: verification and validation
QA: Verification and Validation
\vspace*{1mm}
\item[-]
Infrastructure safety and economy
% All models available for civil engineers
\vspace*{1mm}
\item[-]
\url{http://real-essi.us/}
\end{itemize}
\vspace*{-56mm}
%\begin{figure}[!hbpt]
\begin{flushright}
\hspace*{5mm}
\includegraphics[width=3.0cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Theory_Introduction/tex_works_psfigures_loading_stage-increments-iterations.pdf}
\end{flushright}
%\vspace*{-0.5cm}
%\end{figure}
%
%
%
%
% The Real-ESSI,
% {\underline {\bf Real}}istic
% %{\underline {\bf M}}odeling and
% %{\underline {\bf S}}imulation of
% {M}odeling and
% {S}imulation of
% {\underline {\bf E}}arthquakes,
% {\underline {\bf S}}oils,
% {\underline {\bf S}}tructures and their
% {\underline {\bf I}}nteraction Simulator is a software, hardware and
% documentation system for time domain,
% linear and nonlinear,
% elastic and inelastic,
% deterministic or probabilistic,
% 3D,
% modeling and simulation of:
%
% \begin{itemize}
% %\vspace*{-1mm}
% \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 and dynamics of soil-structure interaction
% % %\vspace*{1mm}
% % % \item[-] dynamics of earthquake-soil-structure system interaction
% % \item[-] dynamics of soil-structure system interaction
% \end{itemize}
%
%
%
% % Used for:
% % \begin{itemize}
% % %\vspace*{1mm}
% % \item[-] Design, linear elastic, load combinations, dimensioning
% %
% %
% % %\vspace*{1mm}
% % \item[-] Assessment, nonlinear/inelastic, safety margins
% % \end{itemize}
% %
%
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Real-ESSI Simulator System}
%
%
% \begin{itemize}
%
%
% \item[] Components
% \begin{itemize}
% \item[-] Real-ESSI Pre (gmsh/gmESSI, X2ESSI)
% \item[-] Real-ESSI Program (local, remote, cloud)
% \item[-] Real-ESSI Post (Paraview/pvESSI, Python, Matlab)
%
% \end{itemize}
%
% \vspace*{1mm}
% \item[] Availability, free executable downloads:
% \begin{itemize}
% % \item[-] Docker Container Images for
% % \begin{itemize}
% % \vspace*{-1mm}
% \item[-] MS-Windows
% \item[-] MacOS
% \item[-] Linux
% % \end{itemize}
%
% % \item[-] Linux Executables
% \item[-] Amazon Web Services
%
% \end{itemize}
%
%
%
%
%
%
% % \begin{itemize}
% % %\vspace*{1mm}
% % \item[-] Educational Institutions: AWS and Linux Executables, free
% % \item[-] Government Agencies, National Labs: AWS GovCloud, free
% % \item[-] Professional Practice: AWS and Linux Executables, commercial
% % %\vspace*{1mm}
% % %%\vspace*{1mm}
% % % \item Sources available to collaborators
% % \end{itemize}
%
%
% %
% % \vspace*{3mm}
% % \item Real-ESSI education and training: theory and applications
% %
%
%
% \vspace*{1mm}
% \item[] Real-ESSI program, documentation, examples:
% \url{http://real-essi.us/}
% %\url{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}
% %
% %\url{http://real-essi.us/}
% %
%
%
% % \vspace*{2mm}
% % \item
% %
%
%
% \end{itemize}
%
%
% \end{frame}
%
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%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Real-ESSI Simulator Quality Assurance}
%
% \begin{itemize}
%
% \item[-]
% Verification available for each element, model, algorithm, ...
%
% \vspace*{10mm}
% \item[-] Validation partially available, working with UCSD, TJU...
%
% % \vspace*{5mm}
% % \item Certification process
% %
% % \begin{itemize}
% %
% % \vspace*{2mm}
% % \item[-] ASME NQA-1
% %
% % \vspace*{2mm}
% % \item[-] ISO-90003-2014
% %
% % \end{itemize}
%
% %\vspace*{3mm}
% %\item[] Verification examples given below
%
% \end{itemize}
%
%
%
%
% \end{frame}
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{Real-ESSI Modeling Features}
%
% \begin{itemize}
%
%
%
% %\vspace*{2mm}
% \item[-] Solids: dry, saturated/liquefaction, elastic, elastic-plastic
%
% %\vspace*{1mm}
% \item[-] Structural elements: beams (B,T), shells, elastic, inelastic
%
% %\vspace*{1mm}
% \item[-] Contact/interface/joint elements: gapping, frictional
% % (EPP, EPH,
% %EPS); Gap/Normal; linear, nonlinear, dry, coupled/saturated,
% % Bonded, Shear/Frictional (EPP, EPH,
% % EPS); Gap/Normal; linear, nonlinear, dry, coupled/saturated,
%
% %\vspace*{1mm}
% \item[-] Super element: stiffness and mass matrices
%
% %\vspace*{1mm}
% \item[-] Material models: soil, rock, concrete, steel...
%
% %\vspace*{1mm}
% \item[-] Seismic input: 1C and 3C, deterministic or probabilistic
%
% %\vspace*{1mm}
% \item[-] Energy calculations: input, el-pl, viscous, algorithmic
%
% %\vspace*{1mm}
% \item[-] Solid/Structure-Fluid interaction, full coupling, OpenFOAM
%
% %\vspace*{1mm}
% \item[-] Forward probabilistic inelastic modeling
%
% %\vspace*{1mm}
% \item[-] Backward probabilistic inelastic modeling: Sensitivities
%
% \item[-] Full Verification and Partial Validation
%
%
% %\vspace*{1mm}
% \item[-] Input: Domain Specific Language
%
% %
% %\vspace*{1mm}
% % \item Modeling features listed at
% % \hspace*{5mm}
% % \href{http://real-essi.us/}{http://real-essi.us/}
% %% \hspace*{5mm}
% %% and
% %% \hspace*{5mm}
% %% \href{http://real-essi.us/}{http://real-essi.us/}
%
%
% \end{itemize}
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Real-ESSI Simulation Features}
%
%
%
% %\vspace*{-10mm}
%
% \begin{itemize}
%
% \item[-] Static loading stages
%
% \vspace*{1mm}
% \item[-] Dynamic loading stages
%
% \vspace*{1mm}
% \item[-] Restart, simulation tree
%
% \vspace*{1mm}
% \item[-] Solution advancement methods/algorithms, \\
% on global and constitutive levels, \\
% with and without enforcing equilibrium
%
% %\vspace*{1mm}
% % \item Load combinations, elastic, for design
%
% \vspace*{1mm}
% \item[-] High Performance Computing (HPC): \\
% Sequential and Parallel
% % % clusters, cloud, supercomputers
% % \begin{itemize}
% % %\vspace*{1mm}
% % \item[.] Fine grained, template mataprograms, small matrix library
% % %\vspace*{1mm}
% % \item[.] Coarse grained, distributed memory parallel
% % \end{itemize}
%
%
% % \vspace*{1mm}
% % \item All Simulation Features are listed at
% % \hspace*{5mm}
% % \href{http://real-essi.us/}{http://real-essi.us/}
% % % \hspace*{5mm}
% % % and
% % % \hspace*{5mm}
% % % \href{http://real-essi.us/}{http://real-essi.us/}
%
%
%
% \end{itemize}
%
%
%
% \vspace*{-60mm}
% %\begin{figure}[!hbpt]
% \begin{flushright}
% \includegraphics[width=2.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Theory_Introduction/tex_works_psfigures_loading_stage-increments-iterations.pdf}
% \end{flushright}
% %\vspace*{-0.5cm}
% %\end{figure}
% %
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% \end{frame}
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Real ESSI Simulator: Domain Specific Language, DSL}
%
% \begin{itemize}
% \item Domain Specific Language (DSL), Yacc \& Lex
% \vspace*{3mm}
% \item English like modeling and simulation language
% \vspace*{3mm}
% \item Parser and compiler, can define functions, models, etc.
% \vspace*{3mm}
% \item Can extend models and methods
% \vspace*{3mm}
% \item Requires units!
% \end{itemize}
% %
% \end{frame}
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% \begin{frame}[fragile]
% \frametitle{DSL: English Language Binding Modeling Parser}
% {\footnotesize
% \begin{lstlisting}
% // Defining variables
% x = 7; // x is double-valued variable with adimenisional units.
% y = 3.972e+2; // Decimal and scientific notation is supported.
% // Operations: All standard arithmetic operations (Unites!)
% a = x + y; // Addition
% b = x - y; // Subtraction
% c = x*y; // Product
% d = x/y; // Quotient
% e = y%x; // Modulus (how many times x fits in y)
% // Predefined variables. For example, the variable 'm' defines 'meter'.
% L1 = 1*m;
% L2 = 40*mm; // Defines L2 to be 40 millimiters.
% L3 = 3.14*cm;
% L4 = 3.14;
% A5 = 3.14*cm^2;
% \end{lstlisting}
% }
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}[fragile]
% \frametitle{DSL: English Language Modeling Parser}
% {\footnotesize
% \begin{lstlisting}
% F1 = 10*kN; // Define few forces.
% F2 = 300*N;
% F3 = 4*kg*g; // Here g is the predefined acceleration
% // due to gravity.
% // Operations are sensitive to units. For example,
% foo = L1 + F1; // Produces an error because units are
% // not compatible. However,
% L4 = L1 + L2 + L3; // is OK.
% // Multiplication (division and modulus) always work
% // because the result produces a quantity with new units
% // (except when the adimensional quantity is involved).
% A = L1*L2;
% pressure = F1 / A;
% // All numbers are converted to SI units (kg - m - s)
% // and internally stored in that system.
% \end{lstlisting}
% }
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{DSL: ESSI Input Language, Basics}
%
%
% \begin{itemize}
%
% \item Angle brackets \lstinline|<>| denotes user input
%
% \item Expected unit (dimension) is given (example:
% \lstinline||, for length unit)
%
% \item Symbol \lstinline|<.>| represents the adimensional quantity.
%
% \item Vertical bar \lstinline+|+ (``OR'' sign)) is used to separate two or more keyword
% options, i.e. \lstinline+[a|b|c]+ is used indicate keyword options
% \lstinline+a+ or \lstinline+b+ or \lstinline+c+.
%
% \item The symbol \lstinline+|...|+ is used to denote where several long options
% exist and are explained elsewhere (an example of this is available below in a
% material model definitions).
%
% \end{itemize}
%
% \end{frame}
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{DSL: ESSI Input Language, Units}
%
%
% All commands require unit consistency. Base units, SI or other (British
% Imperial) can be used
% \begin{itemize}
% \item length, symbol $L$, units [m, in, ft]
% \item mass, symbol $M$, units [kg, lb],
% \item time, symbol $T$, units [s]
% \end{itemize}
%
% Derived units can also be used:
%
% \begin{itemize}
% \item angle, symbol rad (radian), unit [$dimensionless, L/L$]
% \item force, symbol N (Newton), units [$N, kN, MN, M*L/T^2$],
% \item stress, symbol Pa (Pascal), units [$Pa, kPa, MPa, N/L^2, M/L/T^2$]
% \item strain, symbol (no symbol), units [$L/L$]
% \item mass density, symbol (no symbol), units [$M/L^3$]
% \item force density, symbol (no symbol), units [$M/L^2/T^2$]
% \end{itemize}
%
% \end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{DSL: ESSI Input Language, Loading Stages}
%
%
% Start a new loading stage with
%
% \lstinline|new loading stage "loading_stage_name";|
%
% \vspace*{0.5cm}
% Example, starting a new loading stage called {\it "self weight load"}
%
% \lstinline|new loading stage "self weight load";|
%
%
%
%
%
%
% \end{frame}
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
% % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}[fragile]
% \frametitle{DSL: Beam Example, Model}
%
%
%
% \begin{figure}[!h]
% \begin{center}
% \includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Cantilever_Beam.pdf}
% %\caption{8 node brick element}
% %\label{fig:8node_command}
% \end{center}
% \end{figure}
%
%
% %}
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}[fragile]
% \frametitle{Real ESSI DSL Example}
%
%
% \vspace*{-8mm}
% \begin{figure}[!h]
% \begin{flushright}
% \includegraphics[width=4cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Cantilever_Beam.pdf}
% %\caption{8 node brick element}
% %\label{fig:8node_command}
% \end{flushright}
% \end{figure}
%
%
% \vspace*{-4mm}
%
% \begin{lstlisting}
% model name "SmallTestModel";
% new loading stage "First_static";
% // Nodal Coordinates
% add node # 1 at (0*m, 0*m, 0*m) with 6 dofs;
% add node # 2 at (0*m, 0*in, 1000*mm) with 6 dofs;
% add element # 1 type beam_elastic with
% nodes (1, 2) cross_section=1.0*m^2
% elastic_modulus=1.0e5*KN/m^2
% shear_modulus=2.0e4*KN/m^2
% torsion_Jx=2*0.083*m^4
% bending_Iy=0.083*m^4 bending_Iz=0.083*m^4
% mass_density=2500.0*kg/m^3
% xz_plane_vector = (0, -1, 0)
% joint_1_offset = (0.0*m, 0.0*m, 0.0*m)
% joint_2_offset = (0.0*m, 0.0*m, 0.0*m);
% \end{lstlisting}
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}[fragile]
% \frametitle{Real ESSI DSL Example}
%
%
% \begin{lstlisting}
% fix node No 1 dofs all;
% add load # 1 to node # 2 type linear Fy=-9*kN;
% define load factor increment 0.01;
%
% define solver UMFPack;
%
% define convergence test
% Norm_Displacement_Increment
% tolerance = 1e-5
% maximum_iterations = 20
% verbose_level = 4;
%
% define algorithm Newton;
%
% simulate 100 steps using static algorithm;
%
% bye;
% \end{lstlisting}
% \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Real-ESSI Model Development}
%
% \begin{itemize}
%
% \item[-] Pre-Processing, model development gmsh/gmESSI
%
% \vspace*{1mm}
% \item[-] Existing model translation, SASSI$\rightarrow$Real-ESSI
%
%
% \vspace*{1mm}
% \item[-] Self documenting input language
%
% \vspace*{1mm}
% \item[-] Units required for all input variables
%
% \vspace*{1mm}
% \item[-] All variables and constants need to be defined by user
%
%
%
% \vspace*{1mm}
% \item[-] Sophistication level of choice
%
% %\vspace*{1mm}
% % \item[-] Reduce modeling uncertainty
%
% \vspace*{1mm}
% \item[-] Model developed in phases
%
% \vspace*{1mm}
% \item[-] Verify model components
%
%
% \vspace*{1mm}
% \item[-] Build confidence in inelastic modeling
%
% \end{itemize}
%
% \end{frame}
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{ESSI Modeling Phases}
%
%
%
% \begin{figure}[htbp]
% \begin{center}
% \includegraphics[width = 2.3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/soil-structure/overview.png}
% \vspace*{-1mm}
% \\
% \includegraphics[width = 0.35cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/free_field_1D/DRM_1D_motion_3D_just_column.jpg}
% \hspace*{5mm}
% % \includegraphics[width = 0.1cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/free_field_1D/DRM1D_Motion3D.png}
% \includegraphics[width = 2.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/free_field_3D/motion3D_DRM3D_free_field.png}
% \hspace*{5mm}
% % \includegraphics[width = 1cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/soil-foundation/soil_foundation.png}
% % \includegraphics[width = 3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/soil-foundation/slice.png}
% \includegraphics[width = 2.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/soil-foundation/foundation_results.png}
% % \includegraphics[width = 3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/soil-structure/overview.png}
% \\
% \vspace*{-3mm}
% \includegraphics[width = 1.0cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/structure/eigen/structure-only.png}
% \hfill
% \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/structure/eigen/eigen1.png}
% \hfill
% \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/structure/eigen/eigen2.png}
% \hfill
% \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/structure/eigen/eigen3.png}
% \hfill
% \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/structure/eigen/eigen4.png}
% \hfill
% \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/structure/eigen/eigen5.png}
% \hfill
% \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/structure/eigen/eigen6.png}
% \hfill
% % \includegraphics[width = 1.0cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/structure/imposed_motion/structure-only.png}
% %\hfill
% \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/structure/imposed_motion/imposed_motion_results.png}
% % \includegraphics[width = 0.1cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/soil-structure/overview.png}
% \\
% \vspace*{-1mm}
% \includegraphics[width = 6cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/soil-structure/DRM3D_motion3D_structure.png}
% \end{center}
% \end{figure}
%
%
%
% \end{frame}
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Real-ESSI Results Post Processing}
%
% \begin{itemize}
%
%
% \item[-] All output is saved (stress, strain, displacement, energy...)
%
% \vspace*{5mm}
% \item[-] Scripts to plot or extract time histories
%
% \vspace*{5mm}
% \item[-] 3D visualization, Paraview with pvESSI plugin
%
%
%
% \end{itemize}
%
% \end{frame}
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Real-ESSI Core Functionality}
%
% \begin{itemize}
%
%
%
% %\vspace*{2mm}
% \item [-] Inelastic, nonlinear analysis for professional practice
%
% %
% % % \vspace*{3mm}
% % \item Usable models for professional practice
% %
% % %\vspace*{2mm}
% % \item Core functionality needed for nonlinear modeling in professional
% % practice
% % %
% %
%
%
% % %\vspace*{0.3cm}
% % %\vspace*{2mm}
% % \item Hierarchy of modeling capabilities,
% %
% % \begin{itemize}
% %
% % %\vspace*{1mm}
% % \item Linear elastic models, elastic constants, viscous damping
% %
% % %\vspace*{1mm}
% % \item Nonlinear models, core functionality, does not require much
% % material data however, sensitivity study is advised
% %
% % %\vspace*{1mm}
% % \item High sophistication nonlinear models, require material data
% %
% %
% % \end{itemize}
% %
%
%
% \vspace*{3mm}
% \item[-] Low/medium/high sophistication models for ESSI analysis
%
%
% \vspace*{3mm}
% \item[-] Set of suggested modeling and simulation parameters
%
% \vspace*{3mm}
% \item[-] Investigate sensitivity of response to model sophistication
%
% \vspace*{3mm}
% \item[-] Investigate sensitivity of response to model parameters
%
% %
% % \vspace*{1mm}
% % \item[] Accurately follow the flow of seismic energy in a
% % soil structure system
% %
% % \vspace*{1mm}
% % \item[] The goal is to create methodology and numerical tool that is used to
% % predict and inform and not to fit
% %
% %
% %
% % %\vspace*{1mm}
% % % \item[] Directing, in space and time, seismic energy flow in the
% % % soil structure system
% %
%
%
% \end{itemize}
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Real-ESSI Core Functionality Components}
%
% \begin{itemize}
%
%
% \item[-] Structural elements: Truss, Beam, Shell, Super-Element
%
% \vspace*{2mm}
% \item[-] Soil, solids: elastic, $G/G_{max}$
%
% \vspace*{2mm}
% % \item[-] Contacts/interfaces/joints: Bonded, Frictional (EPP, EPH, EPS), Gap
% \item[-] Contacts/interfaces/joints: Bonded, Frictional, Gap
% open/close
%
%
% \vspace*{2mm}
% \item[-] Loads: Static, Dynamic, Earthquake 1C/3$\times$1C/3C, Restart
%
% \vspace*{2mm}
% \item[-] Simulation: explicit/no-equilibrium, Implicit/equilibrium
%
% \vspace*{2mm}
% \item[-] Core Functionality Application programs: APs
% %
%
%
% \end{itemize}
%
%
% \end{frame}
%
%
%
%
%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Real-ESSI Education and Training}
%
% \vspace*{1mm}
% \begin{itemize}
%
%
% % \item[] Real-ESSI Eduction
% % \begin{itemize}
% %\vspace*{1mm}
% \item[-] In-person and online courses
% %\vspace*{1mm}
% % \item[-] Educational short videos
% % \vspace*{1mm}
% % \item[-] Professional practice
% % \vspace*{1mm}
% % \item Developers
%
% %\vspace*{1mm}
% % \item[-] Practical examples available
% % in lecture notes, and documentation
% \vspace*{2mm}
% \item[-] Lecture Notes/Book:
% \begin{itemize}
% \item[(I)] Theory and Computational Formulation,
% \item[(II)] Software and Hardware System,
% \item[(III)] Verification and Validation,
% \item[(IV)] Modeling and Simulation Examples,
% \item[(V)] Application to Practical Engineering Problems.
% \end{itemize}
%
% % \end{itemize}
%
% %\vspace{1mm}
% % \item Documentation, extensive
%
% \vspace*{2mm}
% \item[-] Nonlinear SSI workshop at SMiRT26
%
% \vspace*{2mm}
% \item[-] Nonlinear SSI short course at SMiRT26
%
% \vspace{2mm}
% \item[-] Documentation and Program at \url{http://real-essi.us/}
%
%
% \end{itemize}
%
% \end{frame}
%
%
%
%
%
%
%
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%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Single and Two Phase Systems}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
% \frametitle{FEM Discretization, Dynamic Equilibrium}
%
%
% \begin{itemize}
%
% \item[-] Dynamic equilibrium:
% $\sigma_{ij,j} = f_{i} - \rho \ddot{u}_{i}$
%
%
%
% \end{itemize}
%
%
%
% %\vspace*{-5mm}
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
% \frametitle{FEM Discretization, PVP}
%
%
% \begin{itemize}
%
% \vspace*{1mm}
% \item[-] Principle of virtual displacements: \\
% $
% \int_{V } \sigma_{ij} \;
% \delta \epsilon_{ij} \; dV
% =
% \int_{V }
% \left(
% f^{B}_{i} - \rho \ddot{u}_{i}
% \right)
% \;
% \delta u_{i} \; dV
% +
% \int_{S } f^{S}_{i} \;
% \delta u_{i} \; dS
% $
%
%
%
% \end{itemize}
%
%
%
% %\vspace*{-5mm}
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
% \frametitle{FEM Discretization, Approximation}
%
%
% \begin{itemize}
%
%
% \vspace*{1mm}
% \item[-] Displacement approximation:
% $u_i \approx \hat{u}_{a} = H_{I} \bar{u}_{Ia}$
%
%
%
% \vspace*{6mm}
% \item[-] Strain:
% $
% \epsilon_{ab} \; \approx \; \hat{\epsilon}_{ab}
% =
% \frac{1}{2} \left(
% \left( H_{I,b} \; \bar{u}_{Ia} \right)
% +
% \left( H_{I,a} \; \bar{u}_{Ib} \right)
% \right)
% $
%
%
% \end{itemize}
%
%
%
% %\vspace*{-5mm}
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
% \frametitle{FEM Discretization, Stress-Strain}
%
%
% \begin{itemize}
%
%
% \vspace*{1mm}
% \item[-] Stress-strain relation:
% $
% \Delta \hat{\sigma}_{ab}
% =
% E_{abcd}
% \left( \Delta \hat{\epsilon}_{cd} - \Delta \epsilon_{cd}^{0} \right)
% +
% \Delta \sigma_{ab}^{0}
% $
%
% \end{itemize}
%
%
%
% %\vspace*{-5mm}
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
% \frametitle{FEM Discretization, Stiffness Symmetries}
%
%
% \begin{itemize}
%
%
% \item[-] Minor symmetry of stiffness tensor:
% $E_{abcd} = E_{bacd}= E_{abdc}$
%
%
%
%
% \end{itemize}
%
%
% \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Finite Element Method}
\begin{itemize}
% \vspace*{2mm}
% \item[-] Virtual displacements cannot be zero, they cancel out
%
% \vspace*{2mm}
% \item[-] Single Phase FEM
\item[-] Single Phase FEM:
\hspace*{2mm}
% \\
%Final FEM equations:
%Single phase FEM
$
M_{AacB} \; \ddot{\bar{u}}_{Bc}
+
K_{AacB} \; \bar{u}_{Bc}
=
F_{Aa}
$
%\\ \nonumber
%A,B &=& 1,2,\dots,\mbox{\# of nodes}
%\\ \nonumber
%a,c &=& 1,\dots,\mbox{\# of dimensions (1, 2 or 3)}
% \vspace*{2mm}
\vspace*{4mm}
\item[-] Two phase FEM, u-p-U:
{\footnotesize
% \hspace*{-25mm}
\begin{eqnarray}
\hspace*{-15mm}
\left[ \begin{array}{ccc}
(M_s)_{KijL} & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & (M_f)_{KijL}
\end{array} \right]
\left[ \begin{array}{c}
\ddot{\overline{u}}_{Lj} \\
\ddot{\overline{p}}_N \\
\ddot{\overline{U}}_{Lj}
\end{array} \right]
+
\left[ \begin{array}{ccc}
(C_1)_{KijL} & 0 & -(C_2)_{KijL} \\
0 & 0 & 0 \\
-(C_2)_{LjiK} & 0 & (C_3)_{KijL} \\
\end{array} \right]
\left[ \begin{array}{c}
\dot{\overline{u}}_{Lj} \\
\dot{\overline{p}}_N \\
\dot{\overline{U}}_{Lj}
\end{array} \right]
\nonumber \\
+
\left[ \begin{array}{ccc}
(K^{EP})_{KijL} & -(G_1)_{KiM} & 0 \\
-(G_1)_{LjM} & -P_{MN} & -(G_2)_{LjM} \\
0 & -(G_2)_{KiL} & 0
\end{array} \right]
\left[ \begin{array}{c}
\overline{u}_{Lj} \\
\overline{p}_M \\
\overline{U}_{Lj}
\end{array} \right]
=
\left[ \begin{array}{c}
\overline{f}_{Ki}^{solid} \\
0 \\
\overline{f}_{Ki}^{fluid}
\end{array} \right]
\nonumber
\end{eqnarray}
}
%
%\vspace*{4mm}
% \item[-] Nonlinear FEM residuals, equilibrium:
\vspace*{4mm}
\item[-] Equilibrium:
\hspace*{2mm}
% \\
$R = F_{external} - F_{internal}$
%$
%R_{Q} =
%F_{Q} -
%\left(
%M_{PQ} \; \ddot{\bar{u}}_{P}
%+
%K_{PQ} \; \bar{u}_{P}
%\right)
%$
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Energy Input and Dissipation}
\begin{itemize}
\vspace*{1mm}
\item[] Energy input, forces, loads
\vspace*{4mm}
\item[] Energy dissipation outside SSI domain:
\begin{itemize}
\item[] SSI system oscillation radiation
\item[] Reflected waves radiation
\end{itemize}
\vspace*{2mm}
\item[] Energy dissipation/conversion inside SSI domain:
\begin{itemize}
\vspace*{1mm}
\item[] Inelasticity of soil, interfaces, structure, dissipators
\vspace*{1mm}
\item[] Viscous coupling with internal/pore and external fluids
% % \item[] potential and kinetic energy
% \item[] potential $\leftarrow \! \! \! \! \! \! \rightarrow$ kinetic energy
\vspace*{1mm}
\item[] Energy deflectors, meta-materials
\end{itemize}
\vspace*{2mm}
%\vspace*{1mm}
% \item[] Numerical energy dissipation (numerical damping/production and period errors)
% \item[] Numerical energy dissipation (damping/production)
\item[] Numerical energy dissipation/production
\end{itemize}
%
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{Energy Dissipation}
\begin{itemize}
\vspace*{4mm}
\item[-]
Rate of plastic energy dissipation:
%$\Phi = \sigma_{ij} \dot{\epsilon}_{ij} - \sigma_{ij} \dot{\epsilon}_{ij}^{el} - \rho \dot{\psi}_{pl} \ge 0$
$\Phi = \sigma_{ij} \Delta {\epsilon}_{ij} -
\sigma_{ij} \Delta {\epsilon}_{ij}^{el} -
\rho \Delta {\psi}_{pl} \ge 0$
% %Following the first and second laws of thermodynamics, the equation for plastic
% %energy dissipation in decoupled material models was presented by
% %%\citeN7{Yang2017a}:
% %
% \begin{equation}
% \Phi = \sigma_{ij} \dot{\epsilon}_{ij} - \sigma_{ij} \dot{\epsilon}_{ij}^{el} - \rho \dot{\psi}_{pl} \ge 0
% \label{equation_plastic_dissipation_final}
% \end{equation}
% %
% where $\Phi$ is the rate of plastic energy dissipation per unit volume,
% $\sigma_{ij}$ is the stress tensor, $\epsilon_{ij}$ is the strain tensor,
% $\epsilon_{ij}^{el}$ is the elastic part of the strain tensor, $\rho$ is the
% mass density of the material, ${\psi}_{pl}$ is the plastic free energy per unit
% volume.
%
\vspace*{4mm}
\item[-]
Increment of viscous energy dissipation/damping:
%dissipation $\Delta {D}_V$,
$\Delta {D}_V = C_{ij} \dot{u}_j \Delta {u}_i $
%
% To compute the energy dissipation due to viscous damping, we start with the
% general form of the equation of motion:
% %
% \begin{equation}
% M_{ij} \ddot{u}_j(t) + C_{ij} \dot{u}_j(t) + K_{ij}^{elpl}(t) u_j(t) = f_i(t)
% \label{equation_of_motion}
% \end{equation}
% %
% where $u_j(t)$ is the vector of generalized displacements, $M_{ij}$ is the mass
% matrix, $C_{ij}$ is the damping matrix, $K_{ij}^{elpl}(t)$ is the
% inelastic stiffness matrix that generally evolves with time, $f_i(t)$ is
% the external load vector.
% %
% For linear viscous damping of the Rayleigh type, the damping matrix is
% expressed as
% %
% \begin{equation}
% C_{ij} = a_M M_{ij} + a_{K} K_{ij}^{el}
% \label{equation_damping_matrix}
% \end{equation}
% %
% where $a_M$ and $a_{K}$ are damping constants with units of s$^{-1}$ and s,
% respectively.
% %
%
%
% %
% The incremental form of energy balance for a dynamic system with viscous damping can be expressed as
% %
% \begin{equation}
% \Delta {W}_{Input} = \Delta {E}_K + \Delta {D}_V + \Delta {W}_M
% \label{equation_energy_balance}
% \end{equation}
% %
% The left hand side of Equation~\ref{equation_energy_balance} is the
% increment of external input work
% %
% \begin{equation}
% \Delta {W}_{Input}= f_i \Delta {u}_i
% \label{equation_energy_balance_W_Input}
% \end{equation}
%
%
% %%
% The three terms on the right hand side of Equation~\ref{equation_energy_balance} are
% the increment of kinetic energy $\Delta {E}_K$, the increment of viscous energy
% dissipation $\Delta {D}_V$, and the increment of material work of the system $\Delta {W}_M$
% %
% \begin{equation}
% \begin{aligned}
% \Delta {E}_K &= M_{ij} \ddot{u}_j \Delta u_i \\
% \Delta {D}_V &= C_{ij} \dot{u}_j \Delta {u}_i \\
% \Delta {W}_M &= K_{ij}^{elpl} {u}_j \Delta {u}_i = \Delta {E}_S + \Delta {E}_P + \Delta {D}_P
% \end{aligned}
% \label{equation_energy_balance_components}
% \end{equation}
% %
% Note that the term of material work $W_M$ can be separated into an elastic part
% and a plastic part.
% These two components are known as the elastic strain energy $E_S$ and the plastic
% work of the system, respectively.
% %
% Then, as mentioned in the previous section, plastic work can be further
% decomposed into plastic free energy $E_P$ and plastic energy dissipation $D_P$.
\vspace*{4mm}
\item[-]
Algorithmic, numerical dissipation:
\\
Newmark, Hilber-Hughes-Taylor, Houbolt, Wilson ...
%
% Newmark time integration method
% %\cite{Newmark1959}
% is used for all cases in this study.
% %
% The forward displacements $^{n+1}u_{i}$ and velocities $^{n+1}\dot{u}_{i}$ are
% expressed in terms of their current values and the forward and current values of
% the acceleration
% %
% \begin{equation}
% \begin{aligned}
% ^{n+1}\dot{u}_{i} &= \; ^{n}\dot{u}_{i} + (1 - \gamma) h \; ^{n}\ddot{u}_{i} + \gamma h \; ^{n+1}\ddot{u}_{i} \\
% ^{n+1}u_{i} &= \; ^{n}u_{i} + h \; ^{n}\dot{u}_{i} + (\frac{1}{2} - \beta) h^{2} \; ^{n}\ddot{u}_{i} + \beta h^{2} \; ^{n+1}\ddot{u}_{i}
% \end{aligned}
% \end{equation}
% %
% where $\Delta t$ is the length of each time step, $\gamma$ and $\beta$ are the
% Newmark integration parameters that controls the amount of algorithmic damping in
% the system.
%
%
% %\citeN{Krenk2006}
% gave the incremental form of the energy balance equation
% \ref{equation_energy_balance} over increment $t_n$ to $t_{n+1}$, for Newmark algorithm
% %
% \begin{equation}
% \begin{aligned}
% & \left. \left[
% \frac{1}{2} M_{ij}^{\star} \dot{u}_{i} \dot{u}_{j}
% +
% \frac{1}{2} K_{ij} {u}_i {u}_j
% +
% \left( \beta - \frac{1}{2} \gamma \right) \frac{1}{2} h^{2} M_{ij}^{\star} \ddot{u}_{i} \ddot{u}_{j}
% \right] \right|^{t_{n+1}}_{t_n}
% = \\
% & \quad \quad + \Delta u_{i} \left[ \frac{1}{2} (f_{i}^{n+1} + f_{i}^{n}) + \left( \gamma - \frac{1}{2} \right) \Delta f_{i} \right] \\
% & \quad \quad - \left( \gamma - \frac{1}{2} \right) \left[ K_{ij} \Delta {u}_i \Delta {u}_j + \left( \beta - \frac{1}{2} \gamma \right) h^{2} M_{ij}^{\star} \Delta \ddot{u}_{i} \Delta \ddot{u}_{j} \right] \\
% & \quad \quad - \frac{1}{2} h \left[ h^{-2} C_{ij} \Delta {u}_i \Delta {u}_j + \frac{1}{4} C_{ij} (\dot{u}_{i}^{n+1} + \dot{u}_{i}^{n}) (\dot{u}_{j}^{n+1} + \dot{u}_{j}^{n}) \right] \\
% & \quad \quad + \frac{1}{2} \left( \beta - \frac{1}{2} \gamma \right)^{2} h^{3} C_{ij} \Delta \ddot{u}_{i} \Delta \ddot{u}_{j}
% \end{aligned}
% \label{equation_energy_balance_increment_krenk}
% \end{equation}
% %
% where the equivalent mass matrix $M_{ij}^{\star}$ is defined as
% %
% \begin{equation}
% M_{ij}^{\star} = M_{ij} + \left( \gamma - \frac{1}{2} \right) h C_{ij}
% \end{equation}
% %
%
%
% Rearranging Equation~\ref{equation_energy_balance_increment_krenk} gives the
% explicit expression for the amount of algorithmic energy dissipation over an
% increment
% %
% % \begin{equation}
% \begin{eqnarray}
% & \left[ {E}_K + {D}_V + {W}_M - {W}_{Input} \right]^{t_{n+1}}_{t_n} = \\
% & \quad \quad + \left( \gamma - \frac{1}{2} \right) \Delta f_{i} \Delta u_{i} + \frac{1}{2} \left( \beta - \frac{1}{2} \gamma \right)^{2} h^{3} C_{ij} \Delta \ddot{u}_{i} \Delta \ddot{u}_{j} \\
% & \quad \quad - \left( \gamma - \frac{1}{2} \right) \left[ K_{ij} \Delta {u}_i \Delta {u}_j + \left( \beta - \frac{1}{2} \gamma \right) h^{2} M_{ij}^{\star} \Delta \ddot{u}_{i} \Delta \ddot{u}_{j} \right] \\
% & \quad \quad - \left. \left[ \frac{1}{2} \left( \gamma - \frac{1}{2} \right) h C_{ij} \dot{u}_{i} \dot{u}_{j} + \left( \beta - \frac{1}{2} \gamma \right) \frac{1}{2} h^{2} M_{ij}^{\star} \ddot{u}_{i} \ddot{u}_{j} \right] \right|^{t_{n+1}}_{t_n} \\
% \end{eqnarray}
% %
% When $\gamma = 0.500$ and $\beta=0.250$, all the terms on the right hand side of
% Equation \ref{equation_energy_balance_increment_krenk_new} vanish, thus no
% algorithmic energy dissipation exists.
% %
% For other values of $\gamma$ and $\beta$, algorithmic energy dissipation, or
% even energy production, is observed in the system.
%
%
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Plastic Energy Dissipation}
\vspace*{2mm}
% Single elastic-plastic element under cyclic shear loading
\begin{itemize}
\item[] Plastic work is NOT plastic dissipation !
\item[] Surface area of $F-\Delta$ or $\sigma-\epsilon$ is NOT plastic dissipation !
% \item[] Plastic dissipation always increases
\end{itemize}
%\vspace*{-7mm}
\begin{figure}[!hbpt]
\begin{center}
\hspace*{-5mm}
\includegraphics[width=11.0truecm]{/home/jeremic/tex/works/Thesis/HanYang/Files_06June2017/DOE_Annual_2017/Figures/Dissipation_Material.png}
\end{center}
\end{figure}
\end{frame}
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%
%
% \begin{frame}
% \frametitle{Nonlinear FEM Analysis Process}
%
%
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=3.0cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Theory_Introduction/tex_works_psfigures_loading_stage-increments-iterations.pdf}
% %\caption{\label{loading_stages-increments_iterations} }
% \end{center}
% \vspace*{-0.5cm}
% \end{figure}
% %
% \end{frame}
%
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%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Nonlinear FEM Equilibrium Iterations}
%
%
%
% \begin{itemize}
%
% \item[-] Global, FEM equilibrium iterations, convergence
%
%
% % \item[-] Convergence criteria
% \begin{itemize}
% \item[-] Force, unbalanced, relative, abs. minimum check
% \item[-] Force, unbalanced, average
% \item[-] Force, unbalanced, absolute
% \item[-] Displacement, incremental, relative, abs. minimum check
% \item[-] Displacement, incremental, average
% \item[-] Displacement, incremental, absolute
% \item[-] Energy, incremental, relative, abs. minimum check
% \item[-] Energy, incremental, average
% \item[-] Energy, incremental, absolute
% \end{itemize}
%
%
%
% \vspace*{3mm}
% \item[-] Local, constitutive level equilibrium iterations
%
%
%
% \end{itemize}
%
%
% \end{frame}
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% % %\subsection{Stochastic Modeling}
% % \subsection{Forward Uncertainty Propagation}
% %
% %
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%
%
% %
% %
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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}
%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \end{frame}
%
%
%
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%
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%
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% \begin{frame} \frametitle{{3D Fokker-Planck-Kolmogorov 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 3-D probabilistic stress-strain response
% %--
% %-- \end{itemize}
% %--
% %--
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \end{frame}
%
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% \begin{frame}
%
% \frametitle{FPK Equation}
%
%
%
% \begin{itemize}
%
% \item Advection-diffusion equation
% %
% \begin{equation}
% \nonumber
% \frac{\partial P(\sigma,t)}{\partial t} = -\frac{\partial}{\partial \sigma}\left[N_{(1)}P(\sigma,t)-\frac{\partial}{\partial \sigma}
% \left\{N_{(2)} P(\sigma,t)\right\} \right]
% \end{equation}
%
% %
%
% \item Complete probabilistic description of response
%
%
% \item Solution PDF is second-order exact to covariance of time (exact mean and variance)
%
%
% \item It is deterministic equation in probability density space
%
% \item It is linear PDE in probability density space
% $\rightarrow$ simplifies the numerical solution process
%
% %\vspace*{0.2truecm}
%
% \end{itemize}
%
% %
% % \vspace*{0.5cm}
% % {%
% % \begin{beamercolorbox}{section in head/foot}
% % \usebeamerfont{framesubtitle}\tiny{B. Jeremi\'{c}, K. Sett, and M. L. Kavvas, "Probabilistic
% % Elasto--Plasticity: Formulation in 1--D", \textit{Acta Geotechnica}, Vol. 2, No. 3, 2007, In press (published
% % online in the \textit{Online First} section)}
% % %\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
% % \end{beamercolorbox}%
% % }
%
%
%
% \end{frame}
%
%
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
%
%
% \frametitle{Template Solution of FPK Equation}
%
%
%
% \begin{itemize}
%
%
%
%
% \item FPK diffusion--advection equation is applicable to any material model $\rightarrow$
% only the coefficients $N_{(1)}$ and $N_{(2)}$ are different for different material models
% % %
% % %
%
%
%
% % %
% \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 Elastic-Plastic Response}
%
%
% \begin{figure}[!hbpt]
% \begin{center}
% %\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDF-edited.pdf}
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2012/DOE-LLNL-workshop-27-28-Feb-2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_Contour_PDF-edited.pdf}
% \end{center}
% \end{figure}
%
% \end{frame}
%
%
%
%
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% \begin{frame}
%
% \frametitle{{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_Randomp0-m.pdf}
% \includegraphics[width=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourHighOCR_RandomG_RandomM-m.pdf}
% \hspace*{-10mm}
% \end{center}
% \end{figure}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \end{frame}
% % -- %%%%%%
%
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%
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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{(non-Gaussian, heterogeneous/ non-stationary)}}
% \begin{itemize}
% \item[] Multi-dimensional Hermite Polynomial Chaos (PC) with {known coefficients}
% \end{itemize}
% %\vspace{0.05in}
% \item Output response process
% \begin{itemize}
% \item[] Multi-dimensional Hermite PC with {unknown coefficients}
% \end{itemize}
% % \vspace{0.05in}
% \item Galerkin projection: minimize the error to compute unknown coefficients of response process
% %\vspace{0.05in}
% \item Time integration using Newmark's method
% \begin{itemize}
% \item[] Update coefficients following an elastic-plastic constitutive law at each time step
% \end{itemize}
%
% \end{itemize}
%
% %\scriptsize
% %Note: PC = Polynomial Chaos
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% % \begin{frame}{Discretization of Input Random Process/Field $\beta(x,\theta)$}
% % \begin{center}
% % \includegraphics[scale=0.35]{/home/jeremic/tex/works/Thesis/FangboWang/slides_13Mar2019/Fangbo_slides/figs/PC_KL_explanation.PNG} \\
% % \end{center}
% %
% %
% % \footnotesize{Note: $\beta(x,\theta)$ is an input random process with any
% % marginal distribution, \\ \hspace{21mm} with any covariance structure;} \\
% % \footnotesize{\hspace{8mm} $\gamma(x,\theta)$ is a zero-mean unit-variance Gaussian random process.} \\
% %
% % \end{frame}
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}{Polynomial Chaos Representation}
%
% %\scriptsize{
% Material random field: \\
% %\vspace{-0.3cm}
% %\begin{equation*}
% $D(x, \theta)= \sum_{i=1}^{P1} a_i(x) \Psi_i(\left\{\xi_r(\theta)\right\})$
% %\end{equation*}
%
%
% \vspace{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}
% \frametitle{Forward Uncertainty Propagation}
%
% \begin{itemize}
%
% \vspace*{3mm}
% \item[-] Given uncertain material and uncertain loads
%
% \vspace*{3mm}
% \item[-] Determine uncertain response, $u_i, \dot{u}_i, \ddot{u}_i,
% \epsilon_{ij}, \sigma_{ij}$, PDFs/CDFs
%
% \vspace*{3mm}
% \item[-] Intrusive, analytic development, to circumvent Monte Carlo
% inefficiencies
%
% \end{itemize}
%
%
% \end{frame}
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{{Cam Clay with Random $G$, $M$ and $p_0$}}
%
%
% \begin{figure}[!hbpt]
% \begin{center}
% \hspace*{-15mm}
% \includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_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}{Forward Uncertainty Propagation}
\vspace*{2mm}
Time Domain Stochastic Elastic-Plastic FEM
%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 and random process, 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}{Forward 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}{Forward 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{Backward Uncertainty Propagation, Sensitivities}
\begin{itemize}
\vspace*{1mm}
\item[-]
Given forward uncertain response, PDFs, CDFs...
% \vspace*{1mm}
% \item[-]
% Contributions of uncertain input to forward uncertainties
\vspace*{1mm}
\item[-]
Sensitivity of forward uncertainty to input uncertainties
%
% \vspace*{1mm}
% \item[-]
% 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}
\vspace*{1mm}
\item[-]
The ANalysis Of VAriance representation
% of $f(x)$
(Sobol 2001)
%
% \vspace*{-6mm}
% \begin{small}
% \begin{eqnarray*}
% f(x_1, ... x_n) =
% f_0
% +
% \sum_{i=1}^{n} f_i(x_i) +
% \sum_{1\leq iz)$
%
% \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}
%
% \end{itemize}
%
% \end{frame}
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% % \begin{frame}
% % \frametitle{Intensity Measure (IM)}
% % IM serves as the proxy of damaging ground motions
% %
% % \vspace{0.3cm}
% %
% % \begin{itemize}
% % \item[-] Does a single IM, e.g., $Sa(T_0)$, represent all uncertainty?
% % %% influencing EDP?
% % %\begin{itemize}
% % %\item[--] \small Structure nonlinearity
% % %% \item[--] \small Liquefaction: PGA and duration
% % %\item[--] \small Higher mode response
% % %\end{itemize}
% %
% % \vspace{3mm}
% %
% % \item[-] Practically difficult/contentious to choose
% %
% % % \begin{itemize}
% % % % \item[--] \small 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}
% %
%
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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}
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% \end{itemize}
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% \end{frame}
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% %
% % \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}
% %
% %
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% %
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% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \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}
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%
% %\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
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%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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
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%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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}
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% % \begin{textblock}{15}(0.3, 11.0)
% % \includegraphics[width=0.29\linewidth]{pic/seismic_risk_result_framework.png}
% % \end{textblock}
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% % \begin{tikzpicture}[remember picture, overlay]
% % \draw[line width=1pt, draw=black, rounded corners=4pt, fill=gray!20, fill opacity=1]
% % ([xshift=-25pt,yshift=-52pt]$(pic cs:a) + (0pt,8pt)$) rectangle ([xshift=95pt,yshift=-18pt]$(pic cs:b)+(0pt,-2pt)$);
% % \end{tikzpicture}
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% % \begin{textblock}{15}(-0.1, 9.3)
% % \scriptsize
% % \quad \quad \quad \quad $\lambda(EDP>z)=$
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% % $\quad \sum N_i(M_i, R_i) P(EDP>z|M_i, R_i)$
% % \end{textblock}
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% % \begin{textblock}{15}(1.6, 10.7)
% % \scriptsize{EDP hazard/risk}
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% \begingroup
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% \frametitle{Time Domain Intrusive PSRA Framework}
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%
% \begin{textblock}{15}(0, 4.0)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(0.3, 3.5)
% \scriptsize{Seismic source characterization}
% \end{textblock}
%
% \begin{textblock}{15}(2.9, 5.2)
% \tiny{UCERF3 (2014)}
% \end{textblock}
%
% \begin{textblock}{15}(5.1, 6.5)
% $\Rightarrow$
% \end{textblock}
%
%
% \begin{textblock}{15}(5.8, 3.9)
% \vspace*{1mm}
% \includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
% \end{textblock}
%
%
% \begin{textblock}{15}(7.1, 6.2)
% \scalebox{.9}{\tiny{Fourier spectra}}
% \\
% \vspace*{-0.2cm}
% \scalebox{.9}{\tiny{\hspace{0.14cm} Boore(2003)}}
% \end{textblock}
%
% \begin{textblock}{15}(6.1, 3.5)
% \scriptsize{Stochastic ground motion}
% \end{textblock}
%
% \begin{textblock}{15}(9.9, 6.5)
% $\Rightarrow$
% \end{textblock}
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% \begin{textblock}{15}(10.5, 4.2)
% % \includegraphics[width=0.35\linewidth]{pic/KL_exact_dis_correlation_from_dis.pdf}
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
% \end{textblock}
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% \scriptsize{Uncertainty characterization \\
% \hspace{0.1cm} Hermite polynomial chaos}
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% \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}
%
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% \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}
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% \end{frame}
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% % %\subsection{Illustrative Example}
%
% % % %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% %\frametitle{TDNIPSRA Example Object}
% \frametitle{Example Object}
%
% \begin{textblock}{15}(0.5, 4.0)
% \includegraphics[width=0.47\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/faults_configuration_new.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(7.5, 3.4)
% \includegraphics[width=0.55\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SSC_legend.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(0.8, 11.6)
% \scriptsize
% \begin{itemize}
% \item Fault 1: San Gregorio fault
% \item Fault 2: Calaveras fault
% \item Uncertainty: Segmentation, \\ slip rate, rupture geometry, etc.
% \end{itemize}
% \end{textblock}
%
% \begin{textblock}{15}(8.5, 11.6)
% \scriptsize
% \begin{itemize}
% \item 371 total seismic scenarios
% \item $M \ 5 \sim 5.5$ and $6.5 \sim 7.0$
% \item $R_{jb} \ 20km \sim 40km$
% \end{itemize}
% \end{textblock}
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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 Material Modeling}
%
% %Uncertain 1D shear response
%
%
%
% \begin{figure}[!htbp]
% \centering
% %\subfloat[Uncertain $H_a$]{
% %\hspace{-0.8cm}
% %\includegraphics[width=0.53\textwidth]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/constitutive_relation_uncertainHa_certainCr_MC_verification.pdf}}
% %\subfloat[Uncertain $H_a$ and $C_r$]{
% %\hspace{-0.2cm}
% %\includegraphics[width=0.53\textwidth]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/constitutive_relation_uncertainHa_uncertainCr_MC_verification.pdf}}
% %\vspace{-2mm}
% %\caption{\label{figure_probabilisitc_constitutive_relation} Intrusive probabilistic modeling of 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}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
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\begingroup
\setbeamertemplate{footline}{}
\begin{frame}
\frametitle{Seismic Risk Analysis}
\begin{textblock}{15}(1.9,3.8)
\scriptsize
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/MIDR_PDF_evolution.pdf}
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/PDF_MIDR_combine.pdf}
\end{textblock}
\begin{textblock}{15}(1.9,9.5)
\scriptsize
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/MIDR_distribution_different_floors.pdf}
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/Risk_MIDR.pdf}
\end{textblock}
\begin{textblock}{15}(0.8, 3.8)
\scriptsize Engineering demand parameter (EDP): Maximum 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 Analysis}
%\vspace{-0.5cm}
\vspace*{2mm}
\begin{itemize}
% \item[-] \small Damage measure (DM) defined on multiple EDPs:
% \item[-] \small Damage measure (DM) defined on single EDP:
\item[-] Damage measure defined on single EDP:
\vspace*{-3mm}
%\begin{textblock}{15}(0.7,8.9)
\begin{table}[!htbp]
\small
\resizebox{0.98\hsize}{!}{
\begin{tabular}{ccccccc}
%\hline
\textbf{DM} & MIDR\textgreater{}0.5\% & \textbf{MIDR\textgreater{}1\%} & MIDR\textgreater{}2\% & PFA\textgreater{}0.5${\rm m/s^2}$ & \textbf{PFA\textgreater{}1\boldsymbol{${\rm m/s^2}$}} & PFA\textgreater{}1.5${\rm m/s^2}$ \\
\hline
\textbf{Risk [/yr]} & 6.66$\times 10^{-3}$ & \textbf{3.83\boldsymbol{$\times 10^{-3}$}} & 9.97$\times 10^{-5}$ & 6.65$\times 10^{-3}$ & \textbf{1.92 \boldsymbol{$\times 10^{-3}$}} & 9.45$\times 10^{-5}$ \\
%\hline
\end{tabular}}
\end{table}
%\end{textblock}
\vspace{4mm}
\item[-] Damage measure (DM) defined on multiple EDPs:
% \vspace{2mm}
{\scriptsize $DM: \{\text{MIDR}>1\%\, \cup \,\text{PFA}>1{\rm m/s^2} \}$, seismic risk is \boldsymbol{$4.2 \times 10^{-3}/yr$} }
\vspace{1mm}
{\scriptsize $DM: \{\text{MIDR}>1\%\, \cap \,\text{PFA}>1{\rm m/s^2} \}$, seismic risk is \boldsymbol{$1.71 \times 10^{-3}/yr$}}
\vspace{3mm}
%\vspace{20mm}
\vspace{4mm}
\item[-] \small Seismic risk for DM defined on multiple EDPs can be quite
different from that defined on single EDP
\end{itemize}
\end{frame}
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% \begin{frame}
%
% %\frametitle{Sensitivity Analysis, Backward Propagation}
% \frametitle{Sensitivity Analysis}
%
% %
%
% \begin{itemize}
%
% \vspace*{2mm}
% \item[-] Given forward uncertain response, PDFs, CDFs...
%
% \vspace*{4mm}
% \item[-] Contributions of uncertain input to forward uncertainties
%
% \vspace*{4mm}
% \item[-] Sensitivity of uncertain response to input uncertainties
%
% \vspace*{4mm}
% \item[-] Sobol indices
%
% %\begin{itemize}
% %
% %
% % \vspace*{2mm}
% \vspace*{1mm}
% \item[-] Total variance in PGA, \underline{in this case}, dominated by uncertain ground motions
% \begin{itemize}
% \vspace*{1mm}
% \item[] $49$\% from uncertain rock motions at depth
% \vspace*{1mm}
% \item[] $2$\% from uncertain soil
% \vspace*{1mm}
% \item[] $49$\% from interaction of uncertain rock motions and uncertain soil
%
%
% \end{itemize}
%
% \end{itemize}
%
% % \item
% %\end{itemize}
%
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% \end{frame}
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\begin{frame}
\frametitle{Sensitivity Example: Probabilistic Site Response}
\vspace*{5mm}
\begin{itemize}
% \item[-] Three-layered ground with \\
% uncertain material properties and \\
% uncertain seismic rock motions
\item[-] Uncertain material: \\
uncertain random field, \\
marginally lognormal \\
distribution, \\
exponential correlation \\
length 10m
\vspace*{2mm}
\item[-] Uncertain seismic \\
rock motions: \\
seismic scenario \\
M=7, R=50km \\
% Stochastic Fourier amplitude spectra, \\
% Stochastic Fourier phase derivative
\end{itemize}
\vspace*{-50mm}
\begin{figure}[!hbpt]
\begin{flushright}
\includegraphics[width=5.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Three-Layer-Model.jpg}
\\
\includegraphics[width=5.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Input-motions.jpg}
\end{flushright}
\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}
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%
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% \begin{frame}
%
% \frametitle{Stochastic Material Parameters}
%
% %
%
% \vspace*{5mm}
% Lognormal distributed random field with PC Dim. 3 Order 2
%
%
% \vspace*{-5mm}
% \begin{figure}[!hbpt]
% \begin{center}
% \hspace*{-10mm}
% \includegraphics[width=13cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Stochastic_material_parameters.jpg}
% \end{center}
% \end{figure}
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% \begin{frame}
%
% \frametitle{Stochastic Seismic Motion Development}
%
% \vspace*{5mm}
% \begin{itemize}
% \item[-] UCERF3 (Field et al. 2014)
% \item[-] Stochastic motions (Boore 2003)
% \item[-] Polynomial Chaos Karhunen-Lo{\`e}ve expansion
% \item[-] Probabilistic DRM (Bielak et al. 2003, Wang et al. 2021)
% \end{itemize}
%
% \vspace*{-3mm}
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/UCERF3.pdf}
% \includegraphics[width=4.0cm]{/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}
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% \end{frame}
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% \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,
% %Inter-frequency 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}
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% % \begin{frame}
% %
% % \frametitle{Stochastic Seismic Motions}
% %
% % %
% %
% % %Lognormal distributed random field with PC Dim. 3 Order 2
% %
% %
% %
% % %\vspace*{-50mm}
% % \begin{figure}[!hbpt]
% % \begin{center}
% % \hspace*{-10mm}
% % \includegraphics[width=12.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Stochastic_seismic_motions.jpg}
% % \end{center}
% % \end{figure}
% %
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% % \begin{frame}
% %
% % \frametitle{Stochastic Seismic Motions, Displacement}
% %
% % %
% %
% % %\vspace*{-50mm}
% % \begin{figure}[!hbpt]
% % \begin{center}
% % \hspace*{-10mm}
% % \includegraphics[width=12.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Stochastic-motions-DISP.jpg}
% % \end{center}
% % \end{figure}
% %
% % \end{frame}
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
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% \begin{frame}
%
% \frametitle{Stochastic Seismic Motions, Accelerations}
%
% %
%
% \vspace*{-1mm}
% \begin{figure}[!hbpt]
% \begin{center}
% \hspace*{-10mm}
% \includegraphics[width=12.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Stochastic-motions-ACC.jpg}
% \end{center}
% \end{figure}
%
% \end{frame}
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% % \begin{frame}
% %
% % \frametitle{Stochastic Site Response}
% %
% % %
% %
% %
% % \vspace*{-3mm}
% % \begin{figure}[!hbpt]
% % \begin{center}
% % \hspace*{-10mm}
% % \includegraphics[width=12.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Stochastic-site-response.jpg}
% % \end{center}
% % \end{figure}
% %
% %
% % \vspace*{-2mm}
% % PC dimensions $\xi_1$, - $\xi_3$ are from uncertain soil
% %
% % PC dimensions $\xi_4$, - $\xi_{153}$ are from uncertain rock motions
% %
% % \end{frame}
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Sensitivity of PGA from Uncertain Soil}
%
% %
%
%
% \vspace*{10mm}
% \begin{itemize}
% \item[-] First 10 terms \\
% from soil uncertainty
%
%
%
% \item[-] Total Sobol \\
% sensitivity index \\
% $S_{1-3}^{PC,total} = 0.51$
%
%
% \end{itemize}
%
%
%
% \vspace*{-40mm}
% \begin{small}
% %\begin{center}
% \begin{table}[!htb]
% \hspace*{50mm}
% \begin{tabular}{l l}
% Sobol Index & Value \\ \hline
% $S_{1,123}^{PC}$ & 0,04389 \\
% $S_{1,118}^{PC}$ & 0,02605 \\
% $S_{1,127}^{PC}$ & 0,02370 \\
% $S_{1,100}^{PC}$ & 0,01759 \\
% $S_{1,103}^{PC}$ & 0,01700 \\
% $S_{1,134}^{PC}$ & 0,01680 \\
% $S_{1,141}^{PC}$ & 0,01611 \\
% $S_{1,110}^{PC}$ & 0,01358 \\
% $S_{1,130}^{PC}$ & 0,01303 \\
% $S_{1,132}^{PC}$ & 0,01068 \\
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% \item[-] First 10 terms \\
% from motions uncertainty
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% \item[-] Total Sobol \\
% sensitivity index \\
% $S_{1-153}^{PC,total} = 0.98$
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% Sobol Index & Value \\ \hline
% $S_{1,123}^{PC}$ & 0,04389 \\
% $S_{137}^{PC}$ & 0,03459 \\
% $S_{110}^{PC}$ & 0,03061 \\
% $S_{118}^{PC}$ & 0,02698 \\
% $S_{1,118}^{PC}$ & 0,02605 \\
% $S_{108}^{PC}$ & 0,02482 \\
% $S_{141}^{PC}$ & 0,02373 \\
% $S_{1,127}^{PC}$ & 0,02370 \\
% $S_{1,100}^{PC}$ & 0,01759 \\
% $S_{1,103}^{PC}$ & 0,01700 \\
% . . . & . . . \\ \hline
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% \item[-]
Total variance in PGA, in this particular case (!),
dominated by uncertain ground motions
\begin{itemize}
\vspace*{4mm}
\item[] $49$\% from uncertain rock motions at depth
\vspace*{4mm}
\item[] $2$\% from uncertain soil
\vspace*{4mm}
\item[] $49$\% from interaction of uncertain rock motions and uncertain soil
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% \frametitle{Appropriate Science and Engineering Quotes}
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% % \vspace*{2mm}
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% \vspace*{2mm}
% \item[-] Fran{\c c}ois-Marie Arouet, Voltaire: \\
% "Le doute n'est pas une condition agr{\'e}able,
% mais la certitude est absurde"
%
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% % \vspace*{2mm}
%
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% %
% % \vspace*{2mm}
% % %1895
% % \item[-] William Thomson, Lord Kelvin:
% % "Heavier-than-air flying machines are impossible."
% %
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% % \vspace*{3mm}
% % \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*{4mm}
% \item[-] Theodore Von K{\'a}rm{\'a}n: \\
% "The Scientist studies what is,
% the engineer creates what has never been"
%
%
%
% % \vspace*{2mm}
% % %1943
% % \item[-] Thomas Watson, IBM Chairman:
% % "I think there is a world market for maybe five computers."
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% \item[-] Niklaus Wirth:
% "Software is getting slower more rapidly than hardware becomes faster."
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\frametitle{Summary}
\begin{itemize}
% \item Importance of using proper models correctly (verification,
% validation, level of sophistication)
%
% \item Reduction of modeling uncertainty
%
%
% \vspace*{4mm}
% \item[-] Engineering analysis uncertainties
% \begin{itemize}
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% \item[] Modeling, Epistemic
% \vspace*{1mm}
% \item[] Parametric, Aleatory
% \end{itemize}
%
\vspace*{2mm}
\item[-] Engineering analysis to \underline{predict} and \underline{inform}
\vspace*{2mm}
% \item[-]
% \url{http://real-essi.us/}
% \url{http://real-essi.info/}
\item[-] Engineer needs to know
\vspace*{2mm}
% \item[-]
% \url{http://real-essi.us/}
% \url{http://real-essi.info/}
\item[-] Education and Training is the Key
%\vspace*{4mm}
% \item[-] Real-ESSI Simulator Systems
%% {\includegraphics[width=12mm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/Real_ESSI_in_different_langauges/Real_ESSI_Chinese.jpeg}}
%% \url{http://real-essi.us/}
%
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% \vspace*{2mm}
% \item[-] Real-ESSI Simulator:
% $\rightarrow$ \hspace*{3mm}
% \url{http://real-essi.us/}
% % \url{http://real-essi.info/}
%
% %
% %\vspace*{1mm}
% % \item[-] Sensitivity analysis
\vspace*{2mm}
\item[-] Collaborators: Feng, Yang, Behbehani, Sinha, Wang, Lacoure,
Wang,
Pisan{\'o}, Abell, Tafazzoli, Jie, Preisig, Tasiopoulou, Watanabe, Luo,
Cheng, Yang
\vspace*{2mm}
\item[-] Funding from and collaboration with the
US-DOE,
US-NRC,
US-NSF,
ATC/US-FEMA,
CNSC-CCSN,
CH-ENSI,
UN-IAEA,
%and Shimizu Corp.
is greatly appreciated
%\vspace*{1mm}
% \item[-] Collaborators: Feng, Yang, Wang, Yang.
%\vspace*{4mm}
%\item[]
%Fran{\c c}ois-Marie Arouet, Voltaire: \\
%"Le doute n'est pas une condition agr{\'e}able,
%mais la certitude est absurde" (Voltaire)
% \vspace*{2mm}
% \item[]
% %Niklaus Wirth:
% "Software is getting slower more rapidly than hardware becomes faster."
% (Wirth)
%
% \vspace*{1mm}
% \item[-] Funding from and collaboration with the ATC/US-FEMA, US-DOE, US-NRC, US-NSF,
% CNSC-CCSN, UN-IAEA, and Shimizu Corp. is greatly appreciated,
%
%\vspace*{1mm}
% \item[-]
% \url{http://sokocalo.engr.ucdavis.edu/~jeremic}
%
\end{itemize}
%
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% % \includegraphics[height=1.1truecm]{/home/jeremic/tex/works/Reports/2019/Pine_Flats_Dam_USSD/USSD_Dam_Report_2019/Figures/Model_Mesh_No_Reservior.pdf}
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