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\title[Fully Coupled FEM Analysis]
{Fully Coupled FEM Analysis}
%\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}}
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% affiliation.
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\author[Jeremi{\'c} et al.] % (optional, use only with lots of authors)
%{Boris~Jeremi{\'c}}
{Boris Jeremi{\'c} \\
%\institute[Computational Geomechanics Group \hspace*{0.3truecm}
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%%{ Professor, University of California, Davis\\
{\small University of California, Davis
\\
~
\\
~
\\
Tongji University \\
Liquefaction Workshop
% \\
%Underground Structures in Liquefiable Ground
}
% % 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 01Nov2023
% \\ ~
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% ~ \vspace*{8mm}
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\subject{}
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% Structuring a talk is a difficult task and the following structure
% may not be suitable. Here are some rules that apply for this
% solution:
% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
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% - 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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% % Introduction to Seismic Ground Motions. Participants will learn
% % about and develop free field seismic ground motions from:
% %
% %
% Ground motions from given surface motions.
%
% Ground motions from small and regional scale models.
%
% Ground Motions for ESSI Modeling. Participants will learn
% about, apply and propagate ground motions through the soil-structure system:
%
% Input ground motion into FEM model,
%
% Application of ESSI ground motions.
%
% Develop hierarchy of input ground motions: 1D, 3x1D and 3D.
%
% Control ground motion frequency content.
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%\section{Inelastic, Coupled Analysis}
\section{Introduction}
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%\subsection*{Real-ESSI Simulator System}
%\subsection{Analysis Features}
\subsection{\ }
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\begin{frame}
\frametitle{Engineering Analysis}
\begin{itemize}
\item[-] Improve analysis and design for infrastructure
\vspace*{3mm}
\item[-] All ESSI problems are three dimensional, 3D
\vspace*{3mm}
\item[-] Simplifying assumptions $\rightarrow$ epistemic uncertainty
%\vspace*{3mm}
% \item[-]
\vspace*{3mm}
\item[-] Design and assessment of infrastructure
\vspace*{3mm}
\item[-] Predict and inform!
\vspace*{3mm}
\item[-] Engineer needs to know!
\end{itemize}
\end{frame}
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% \begin{frame}
% \frametitle{Long Lasting Infrastructure}
%
%
%
% \begin{itemize}
%
%
% %\vspace*{0.3cm}
% \item[]
%
%
% \begin{figure}[!hbpt]
% \begin{center}
% %
% %\hspace*{-10mm}
% \includegraphics[height=2.8truecm]{/home/jeremic/tex/works/Conferences/2021/ASCE-4_Kennedy_Lecture/present/Aya-Sofia_03_1990.jpg}
% %\includegraphics[height=2.8truecm]{/home/jeremic/tex/works/Conferences/2021/CU-Boulder-GEGM-seminar-series-02Apr2021/present/Saint_Sophia_Constantinopolis.jpg}
% %\hfill
% \hspace*{10mm}
% \includegraphics[height=2.8truecm]{/home/jeremic/tex/works/Conferences/2021/ASCE-4_Kennedy_Lecture/present/ZhaozhouBridge.jpg}
% \\
% \vspace*{2mm}
% \includegraphics[height=3.2truecm]{/home/jeremic/tex/works/Conferences/2021/ASCE-4_Kennedy_Lecture/present/Four_Water_Wheels_Hama_Syria.jpg}
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% \includegraphics[height=3.2truecm]{/home/jeremic/tex/works/Conferences/2022/USBR_V_and_V_Mar-Apr-May_2022/present/Lecture01_Modeling/Hoover_dam.jpg}
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% % \includegraphics[height=1.3truecm]{/home/jeremic/tex/works/Conferences/2022/USBR_V_and_V_Mar-Apr-May_2022/present/Lecture01_Modeling/Kuca_Bajina_Basta-Drina.jpg}
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% \end{itemize}
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\begin{frame}
\frametitle{Prediction under Uncertainty}
\begin{itemize}
%\vspace*{1mm}
% \item \underline{Modeling, Epistemic Uncertainties},
\item[-] Modeling, Epistemic Uncertainties
\begin{itemize}
\vspace*{1mm}
\item[] Modeling simplifications
\vspace*{1mm}
\item[] Low, medium, high sophistication modeling and simulation
\vspace*{1mm}
\item[] Modeling sophistication level for confidence in results
\vspace*{1mm}
\item[] Verification and Validation
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{2mm}
% \item \underline{Parametric, Aleatory Uncertainties},
\item[-] Parametric, Aleatory Uncertainties
\begin{itemize}
\vspace*{1mm}
\item[] ${M} \ddot{u_i} + {C} \dot{u_i} + {K}^{ep} {u_i} = {F(t)}$
\vspace*{1mm}
\item[] Uncertain: mass $M$, viscous damping $C$ and stiffness $K^{ep}$
\vspace*{1mm}
\item[] Uncertain loads, $F(t)$
\vspace*{1mm}
\item[] Results are PDFs and CDFs for $\sigma_{ij}$, $\epsilon_{ij}$, $u_i$, $\dot{u}_i$, $\ddot{u}_i$
\end{itemize}
\end{itemize}
%
%
% %Le doute n'est pas un {\'e}tat bien agr{\'e}able,\\
% mais l'assurance est un {\'e}tat ridicule. (Fran{\c c}ois-Marie Arouet, Voltaire)
\end{frame}
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%
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\begin{frame}
\frametitle{Engineer, Analyst}
\begin{itemize}
%\vspace*{0.3cm}
\item[-] Educated, Trained, Experienced
%\vspace*{1mm}
% \item[-] From simple to sophisticated modeling
%\vspace*{2mm}
% \item[-] Simplified analysis
%
\vspace*{2mm}
\item[-] Sound engineering judgement
\vspace*{2mm}
\item[-] Assess various analysis sophistication levels
% %
% \vspace*{2mm}
% \item[-] Expert engineering analyst
%
% \vspace*{1mm}
% \item[-] Credibility of developed models
% %
% %
% %
% \vspace*{1mm}
% \item[-] Confidence in and credibility of analysis results
% %
%
\vspace*{2mm}
\item[-] Engineer is in full control of the model and the analysis
\vspace*{2mm}
\item[-] Engineer uses models to assess design options
%
\vspace*{2mm}
\item[-] Confidence in all modeling choices
\vspace*{2mm}
\item[-] Confidence in all analysis results
% %\vspace*{2mm}
% % \item[-] Confidence in all model components
%
% \vspace*{2mm}
% \item[-] Confidence in all analysis results
\end{itemize}
\end{frame}
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Expert Analyst, Engineer}
%
%
%
% \begin{figure}[!h]
% \begin{center}
% %\vspace*{-2.5cm}
% %\vspace*{-1.0cm}
% %\hspace*{-1.5cm}
% {\includegraphics[height=3.0cm]{/home/jeremic/tex/works/Conferences/2022/IFNFEES02_01-02Jul2022/present/Hartford-Coliseum-failure-01.jpg}}
% \hfill
% {\includegraphics[height=3.0cm]{/home/jeremic/tex/works/Conferences/2022/IFNFEES02_01-02Jul2022/present/Hartford-Coliseum-failure-03.jpg}}
% \hfill
% {\includegraphics[height=3.0cm]{/home/jeremic/tex/works/Conferences/2022/IFNFEES02_01-02Jul2022/present/Hartford-Coliseum-failure-02.jpg}}
% %\hspace*{-1.5cm}
% %\vspace*{-1.5cm}
% %\vspace*{-5.0cm}
% \end{center}
% \end{figure}
%
% {\tiny Hartford Coliseum Collapse, 1978 (Martin and Delatte, ASCE-JPCF (2001))}
%
% % \begin{itemize}
% %
% %
% %
% % \end{itemize}
%
% \end{frame}
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\begin{frame}
\frametitle{Analysis Program}
\begin{itemize}
%\vspace*{0.3cm}
\item[-] Hierarchy of model sophistication capabilities
% \item[-] Hierarchy, levels of modeling capabilities
\vspace*{3mm}
\item[-] Hierarchy of simulation/algorithmic capabilities
%
\vspace*{3mm}
\item[-] Full Verification !
\vspace*{3mm}
\item[-] Extensive Validation
\vspace*{3mm}
\item[-] Confidence in analysis results
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Modeling Features}
\begin{itemize}
\vspace*{2mm}
\item[-] Solid elements: dry, (un-)saturated, elastic, inelastic
\vspace*{1mm}
\item[-] Structural elements: beams, shells, elastic, inelastic
\vspace*{1mm}
\item[-] Contact/interface/joint elements: Bonded, Shear/Frictional;
% (EPP, EPH, EPS);
Gap/Normal; linear, nonlinear, dry, coupled/saturated,
%\vspace*{1mm}
% \item[-] Super element: stiffness and mass matrices
\vspace*{1mm}
\item[-] Material models: soil, rock, concrete, steel...
\vspace*{1mm}
\item[-] Seismic input: 1C and 3C, deterministic or probabilistic
\vspace*{1mm}
\item[-] Energy dissipation: elastic-plastic, viscous, algorithmic
\vspace*{1mm}
\item[-] Solid/Structure-Fluid interaction, Internal and External
\vspace*{1mm}
\item[-] Intrusive probabilistic inelastic modeling, Forw. and Backw.
%
%\vspace*{1mm}
% \item Modeling features listed at
% \hspace*{5mm}
% \href{http://real-essi.info/}{http://real-essi.info/}
%% \hspace*{5mm}
%% and
%% \hspace*{5mm}
%% \href{http://real-essi.info/}{http://real-essi.info/}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Simulation Features}
%\vspace*{-10mm}
\begin{itemize}
\item[-] Static loading stages
\vspace*{1mm}
\item[-] Dynamic loading stages
\vspace*{1mm}
\item[-] Explicit and Implicit computations
\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
% 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.info/}{http://real-essi.info/}
% % \hspace*{5mm}
% % and
% % \hspace*{5mm}
% % \href{http://real-essi.info/}{http://real-essi.info/}
\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}
%
\end{frame}
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%\subsection{Verification and Validation}
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% -
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - \begin{frame}
% - \frametitle{Verification and Validation}
% - \begin{itemize}
% -
% -
% -
% -
% -
% - \item[-] {{ Verification:} provides evidence that the model is solved
% - correctly.} Mathematics issue. Well developed
% - for the Real ESSI Simulator.
% -
% - \vspace*{4mm}
% - \item[-] {{ Validation:} provides evidence that the correct model is
% - solved.} Physics issue. Work in progress.
% -
% - \vspace*{4mm}
% - \item[-] { Prediction:} use of computational model
% - to foretell the state of a physical system under consideration under
% - conditions for which the computational model has not been validated.
% -
% - % \item { Prediction under Uncertainty}: use of computational model
% - % to predict the state of SSI system under
% - % conditions for which the computational model has not been validated.
% - %
% - %
% - %
% - % \vspace*{1mm}
% - % \item Modeling and parametric uncertainties are always present, need to be
% - % addressed
% - %
% - % %\vspace*{1mm}
% - % % \item Predictive capabilities with {low Kolmogorov Complexity}
% - % %
% - %
% - % \vspace*{1mm}
% - % \item Goal: Predict and Inform and rather than (force) Fit
% - %
% - %
% -
% -
% -
% - \end{itemize}
% - \end{frame}
% -
% -
% -
% -
% - % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - % \begin{frame}
% - % \frametitle{V \& V Motivation}
% - %
% - %
% - %
% - % \begin{itemize}
% - %
% - %
% - % %\vspace*{0.5cm}
% - % \item How much can (should) we trust model implementations (verification)?
% - %
% - % %\vspace*{0.5cm}
% - % \item How much can (should) we trust numerical simulations (validation)?
% - %
% - % %\vspace*{0.5cm}
% - % \item How good are our numerical predictions?
% - %
% - % %\vspace*{0.5cm}
% - % \item Can simulation tools be used for improving safety and economy?
% - %
% - % % \vspace*{2.0truecm}
% - % \item V \& V procedures are the primary means of assessing accuracy in
% - % modeling and computational simulations
% - %
% - % % \vspace*{1.0truecm}
% - % \item V \& V procedures are the tools with which we build confidence and
% - % credibility in modeling and computational simulations
% - %
% - %
% - %
% - %
% - % %\vspace*{2.5cm}
% - % %\item How do we use experimental simulations to improve models
% - %
% - %
% - %
% - %
% - % \end{itemize}
% - %
% - %
% - %
% - %
% - % \end{frame}
% - %
% - %
% - %
% -
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -
% -
% - % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - % \begin{frame}
% - % \frametitle{Role of Verification and Validation}
% - %
% - %
% - %
% - % \begin{figure}[!h]
% - % \begin{center}
% - % \hspace*{-2cm}
% - % %{\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Conferences/2012/ASME_V_and_V_symposium/presentetation/RoleVV_NEW_knowledge.pdf}}
% - % {\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/RoleVV_NEW_knowledge.pdf}}
% - % %{\includegraphics[width=6.5cm]{/home/jeremic/tex/works/Conferences/2011/USNCCM11_Minneapolis/Coupled/Present/VandV_ODEN.jpg}}
% - % {\includegraphics[width=6.5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/VandV_ODEN.jpg}}
% - % \hspace*{-2cm}
% - % \end{center}
% - % \end{figure}
% - %
% - % {Oberkampf et al. \hspace*{4cm} Oden et al.}
% - % %
% - % %\item Models available (some now, some later)
% - % %\vspace*{-2.0cm}
% - %
% - % %
% - % %\item Models available (some now, some later)
% - % %\vspace*{-2.0cm}
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% -
% -
% -
% - \begin{frame}
% - \frametitle{Fundamentals of Verification and Validation}
% -
% -
% -
% - %--
% -
% -
% - \begin{figure}[!h]
% - % \vspace*{-0.5cm}
% - \hspace*{-8mm}
% - %\begin{center}
% - %{\includegraphics[width=11cm]{/home/jeremic/tex/works/Conferences/2005/OpenSeesWorkshopAugust/DeveloperSymposium/VerifValidFund01.pdf}}
% - {\includegraphics[width=12cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/VerifValidFund01.pdf}}
% - %\end{center}
% - \end{figure}
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% - \end{frame}
% -
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% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - \begin{frame}
% - \frametitle{Important Sources}
% -
% -
% -
% - \begin{itemize}
% -
% -
% -
% -
% - %\item {\it Short Course on Verification and Validation in
% - %Computational Mechanics}, by {\sc Dr. William Oberkampf}, Sandia National Laboratories
% - %July 27th, 2003, Albuquerque, New Mexico.
% -
% - \item
% - {\sc W.~L. Oberkampf, T.~G. Trucano, and C. Hirsch.}
% - Verification, validation and predictive capability in computational
% - engineering and physics.
% - In {\em Proceedings of the Foundations for Verification and
% - Validation on the 21st Century Workshop}, pages 1--74, Laurel, Maryland,
% - October 22-23 2002. Johns Hopkins University / Applied Physics Laboratory.
% - %{\href{http://sokocalo.engr.ucdavis.edu/~jeremic/UsefulReadings/Oberkampf-Trucano-Hirsch.pdf}
% - %{PDF available here}}
% -
% -
% - \item
% - {\sc P.~J. Roache.}
% - {\em Verification and Validation in Computational Science and
% - Engineering}.
% - Hermosa publishers, 1998.
% - ISBN 0-913478-08-3.
% - %
% -
% -
% - % \item Material from {\it Verification and Validation in Computational Mechanics}
% - % web site \texttt{http://www.usacm.org/vnvcsm/} at the USACM.
% -
% - \item William L. Oberkampf and Christopher J. Roy. Verification and Validation
% - in Scientific Computing. Cambridge University Press, 2010.
% -
% -
% -
% - \end{itemize}
% -
% -
% -
% -
% - \end{frame}
% -
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% - %
% - %
% -
% -
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% -
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% -
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% -
% -
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% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - \begin{frame}
% - \frametitle{Verification}
% -
% - \begin{itemize}
% -
% - \item Source code management
% - \item Source code verification
% -
% - \item Constitutive integration
% - \item Static and dynamic behavior of single phase solids
% - \item Static and dynamic behavior of fully and partially saturated, fully
% - coupled, porous solid-pore fluid problems
% - \item Static and dynamic behavior of structural elements
% - \item Static and dynamic behavior of special elements
% - (contacts-interface/gap-frictional/dry-saturated, isolators/dissipators)
% - \item Static and dynamic FEM solution advancement
% - \item Seismic wave propagation problems
% - \item FEM Model verification, hierarchy of models
% - %\item Static and Dynamic Behavior of Soil-Structure-Interaction
% -
% - \end{itemize}
% -
% - \end{frame}
% -
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% \begin{frame}
% \frametitle{Constitutive Integration Verification}
%
%
% \begin{itemize}
%
% \item Asymptotic regime of convergence
%
% \item Richardson extrapolation
%
% \item Grid convergence index
%
% \end{itemize}
%
%
% \begin{center}
% \begin{figure}[!htbp]
% \vspace*{-3mm}
% \includegraphics[width=5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/asymptotic_regime.pdf}
% \includegraphics[width=3.5truecm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/asymptotic_nonassociate.pdf}
% \\
% \includegraphics[width=3.0truecm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/error-map/DPAF_implicitErrMap.pdf}
% \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/GCI/NonAssociate.pdf}
% \vspace*{-3mm}
% \end{figure}
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% \begin{frame}
% \frametitle{Seismic Energy Input and Dissipation}
%
% \begin{itemize}
%
% \vspace*{1mm}
% \item[] Seismic energy input, through a closed boundary (DRM)
%
%
% \vspace*{4mm}
% \item[] Mechanical dissipation outside SSI domain:
% \begin{itemize}
% \item[] Reflected wave radiation
% \item[] SSI system oscillation radiation
% \end{itemize}
% %\vspace*{1mm}
% \item[] Mechanical dissipation/conversion inside SSI domain:
% \begin{itemize}
% \item[] Inelasticity of soil and contact zone
% \item[] Inelasticity/damage of structure and foundation
% \item[] Viscous coupling of fluids and soils and structure
% % % \item[] potential and kinetic energy
% % \item[] potential $\leftarrow \! \! \! \! \! \! \rightarrow$ kinetic energy
% \end{itemize}
%
%
%
% %\vspace*{1mm}
% % \item[] Numerical energy dissipation (numerical damping/production and period errors)
% % \item[] Numerical energy dissipation (damping/production)
% \item[] Numerical energy dissipation/production
%
%
% \end{itemize}
%
% %
% \end{frame}
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
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% \begin{frame}
%
% \frametitle{Energy Dissipation Verification: \\ Plastic Work $\neq$ Plastic Dissipation}
%
% %\begin{itemize}
% %\item Negative incremental energy dissipation
% %\item Plastic work is NOT plastic dissipation
% %\end{itemize}
%
% \begin{figure}[!H]
% \begin{center}
% \includegraphics[height=4.5cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_06June2017/DOE_Annual_2017/Figures/Negative_Dissipation_Problem.png}
% \end{center}
% \end{figure}
%
% \vspace*{-2mm}
% \begin{itemize}
%
% \item[] Direct violation of the second law of thermodynamics
%
% %\vspace*{3mm}
% \item[] 600 papers since 1990 (!?!) repeat this error
%
% % \vspace*{3mm}
% % \item[] Important form of energy missing: {Plastic Free Energy}
% %
% % \vspace*{3mm}
% % \item[] First described by Taylor and Quinney in 1934!
% %
% % \vspace*{3mm}
% % \item[] Plastic Work vs. {Plastic Energy Dissipation}
%
%
% %\vspace*{4mm}
% %\item[] However it seems to be forgotten
%
% \end{itemize}
%
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Dynamic Time Stepping Verification}
%
% %\footnotesize
% Based on the amplification matrix ${\bf A}$, to calculate the analytical solution of
% damping ratios and period shift.
%
% Example: Hilber-Hughes-Taylor $\alpha=-0.1$
%
% \begin{figure}
% \hspace*{-5mm}
% \includegraphics[width=5.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/dynamic/hht/HHT-alpha-01-xi.jpg}
% \includegraphics[width=5.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/dynamic/hht/HHT-alpha-01-period-shift.jpg}
% \hspace*{-5mm}
% \end{figure}
% % HHT Integration $\alpha=-0.2$
% % \begin{figure}
% % \includegraphics[width=0.36\textwidth]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/dynamic/hht/HHT-alpha-02-xi.jpg}
% % \includegraphics[width=0.36\textwidth]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/dynamic/hht/HHT-alpha-02-period-shift.jpg}
% % \end{figure}
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%
% \end{frame}
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% \begin{frame}
% \frametitle{Seismic Input Verification, DRM}
% % \begin{figure}
% % % \includegraphics[scale=0.35]{Present06_figs/3DModel.eps}
% % \includegraphics[width=3cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/3DModel.eps}
% % \end{figure}
%
% \begin{figure}
% %\hspace*{-5mm}
% \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/90m_Northridge_EW.eps}
% \\
% \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/90m_Northridge_NS.eps}
% \\
% \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/90m_Northridge_UD.eps}
% \end{figure}
%
%
%
%
% % local
% % local% local
% % local%\vspace*{-2mm}
% % local\begin{center}
% % local\hspace*{-16mm}
% % local%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
% % local\movie[label=show3,width=61mm,poster,showcontrols]
% % local {\includegraphics[width=60mm]
% % local {movie_ff_3d_mp4_icon.jpeg}}
% % local {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_ff_3d.mp4}
% % local%\hspace*{-2mm}
% % local%\hfill
% % local%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
% % local\movie[label=show3,width=61mm,poster,showcontrols]
% % local {\includegraphics[width=60mm]{movie_ff_1d_mp4_icon.jpeg}}
% % local {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_ff_1d.mp4}
% % local\hspace*{-16mm}
% % local\end{center}
% % local% local
%
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% % online % online
% % online % online
% % online % online
% % online \begin{center}
% % online \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_3d.mp4}
% % online {\includegraphics[width=50mm]{movie_ff_3d_mp4_icon.jpeg}}
% % online %
% % online \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_1d.mp4}
% % online {\includegraphics[width=50mm]{movie_ff_1d_mp4_icon.jpeg}}
% % online \end{center}
% % online % online
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% % online
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% \begin{frame}
% \frametitle{Verification: ANDES Shell}
%
% \begin{center}
% % \includegraphics[width=0.6\textwidth]{./_Jose_Files/fig1_static_tests.pdf}
% \hspace*{-10mm}
% \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/fig1_static_tests.pdf}
% % \includegraphics[width=0.6\textwidth]{./_Jose_Files/fig2_static_tests.pdf}
% \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/fig2_static_tests.pdf}
% \hspace*{-10mm}
% \end{center}
%
% \begin{flushleft}
% \begin{tabular}{|l|l|}
% \hline
% $N_{\texttt{subd}}$ & $u_z$ \\ \hline
% 2 & 96.2118 \\
% 7 & 100.096 \\
% 101 & 100.002 \\
% & \\ \hline
% \end{tabular}
% \end{flushleft}
%
%
% \vspace*{-30mm}
%
% \begin{flushright}
% % \includegraphics[width=0.9\textwidth]{./_Jose_Files/Test_shell_andes_1_free_vibration_EigenMode1.png}
% %\hspace*{-10mm}
% \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/Test_shell_andes_1_free_vibration_EigenMode1.png}
% \\
% \vspace*{-8mm}
% \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/Test_shell_andes_2_free_vibration_EigenMode1.png}
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% \end{flushright}
%
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
% \begin{frame}
% \frametitle{Verification: Irregular Solids and Poisson's Ratio}
%
%
% % \begin{itemize}
% % \item Irregular Mesh
% % \item Mesh Refinement
% % \end{itemize}
%
% \begin{figure}
% % \hspace*{-25mm}
% % \includegraphics[width=5.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/beam/beam_8brick.pdf}
% % % \hspace*{-25mm}
% % % \includegraphics[width=5.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/beam/beam_8brick_more_2.pdf}
% \hspace*{-25mm}
% \includegraphics[width=6.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/beam/beam_brick27_shape1_vertical.pdf}
% \hspace*{-35mm}
% \includegraphics[width=6.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/beam/beam_brick27_shape2_vertical.pdf}
% \hspace*{-35mm}
% \includegraphics[width=6.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/beam/beam_brick27_shape3_vertical.pdf}
% \hspace*{-25mm}
% \end{figure}
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% \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/beam/irregular_shape.png}
% \end{figure}
%
% % \begin{itemize}
% % \item Verification Results of High Poisson's ratios
% % \end{itemize}
%
% \vspace*{-5mm}
%
% \begin{figure}
% \hspace*{-9mm}
% \includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/beam/error27brick_beam_different_poisson_ratio_disp_div100.jpeg}
% \hspace*{3mm}
% \includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/beam/high_poisson.png}
% \end{figure}
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% \begin{frame}
% \frametitle{Verification of Solid Shell/Plate}
%
%
% \begin{figure}
% \hspace*{-9mm}
% \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/beam/circular_plate1.pdf}
% \hspace*{-5mm}
% \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/beam/circular_plate3.pdf}
% \hspace*{-5mm}
% \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/beam/circular_plate4.pdf}
% \hspace*{-5mm}
% \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/beam/circular_plate6.pdf}
% \hspace*{-9mm}
% \end{figure}
%
% \vspace*{-2mm}
%
% \begin{itemize}
% \item Simply supported and clamped ends
% \item Timoshenko's analytic solutions
% \end{itemize}
%
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% \vspace*{-3mm}
%
% \begin{figure}
% \hspace*{-2mm}
% \includegraphics[width=3.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/beam/plate_error_plot.jpeg}
% \hspace*{3mm}
% \includegraphics[width=3.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/beam/plate_error_plot_clamped.jpeg}
% \end{figure}
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% \frametitle{Verification of Boussinesq Problem}
%
%
% \begin{figure}
% \hspace*{-9mm}
% \includegraphics[width=5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/boussinesq/point_load_on_half_space.png}
% \hspace*{3mm}
% \includegraphics[width=2.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/boussinesq/model3Dview.JPG}
% \end{figure}
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% \vspace*{-3mm}
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% \hspace*{-9mm}
% \includegraphics[width=4.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/boussinesq/Boussinesq-8NodeBrick.jpeg}
% \hspace*{5mm}
% \includegraphics[width=4.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/boussinesq/Boussinesq-27NodeBrick.jpeg}
% \end{figure}
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% \frametitle{Verification for Fully Coupled Problems}
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% \includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2011/Verification_Coupled_Systems/FIGS/Gajo_comp.pdf}
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% \frametitle{Wave Propagation, Mesh Size Effects}
% \begin{center}
% \begin{figure}
% \includegraphics[scale=0.40]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present03_img/Vs1000_8Hz_10m.eps}
% \end{figure}
% {\small (Case 1, Vs = 1000 m/s, Cutoff Fq. = 8 Hz, E. Size = 10 m)}
% \end{center}
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% \frametitle{Wave Propagation, Mesh Size Effects}
% \begin{center}
% \begin{figure}
% \includegraphics[scale=0.40]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present03_img/Vs1000_8Hz_20m.eps}
% \end{figure}
% {\small (Case 1, Vs = 1000 m/s, Cutoff Fq. = 8 Hz, E. Size = 20 m)}
% \end{center}
% \end{frame}
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% % \begin{frame}
% % \frametitle{Mesh Size Effects Analysis Results}
% % \begin{center}
% % \begin{figure}
% % \includegraphics[scale=0.40]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present03_img/Vs1000_15Hz_10m.eps}
% % \end{figure}
% % {\small (Case 1, Vs = 1000 m/s, Cutoff Fq. = 15 Hz, E. Size = 10 m)}
% % \end{center}
% % \end{frame}
% %
% %
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% % \frametitle{Mesh Size Effects Analysis Results}
% % \begin{center}
% % \begin{figure}
% % \includegraphics[scale=0.40]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present03_img/Vs1000_15Hz_20m.eps}
% % \end{figure}
% % {\small (Case 1, Vs = 1000 m/s, Cutoff Fq. = 15 Hz, E. Size = 20 m)}
% % \end{center}
% % \end{frame}
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\section{Inelastic, Coupled Analysis}
%
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\subsection{Numerical Formulation}
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\begin{frame}
\frametitle{FEM Formulation Assumptions}
\begin{itemize}
\item[-] 3D Geometry
\vspace*{1mm}
\item[-] Solids and Structures
\vspace*{1mm}
\item[-] Material nonlinear, inelastic and/or elastic
\vspace*{1mm}
\item[-] Small deformations:
\begin{itemize}
\item[] small strain
\item[] small translations
\item[] small rotations
\end{itemize}
%\vspace*{1mm}
% \item[-] Dry, single phase material
\vspace*{1mm}
\item[-] Fully or Partially/Un- Saturated, two phase material
\vspace*{1mm}
\item[-] Mixture of pore fluid and porous solid
\vspace*{1mm}
\item[-] Compressible pore fluid and compressible porous solid
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{Two Phase, Fully Coupled Systems}
\begin{itemize}
\item[-] Based on: \\
O. C. Zienkiewicz and T. Shiomi. Dynamic behaviour of saturated porous media;
the generalized Biot formulation and its numerical solution. International
Journal for Numerical and Analytical Methods in Geomechanics, 8:71-96, 1984.
\vspace*{4mm}
\item[-] Full coupling of internal, pore fluid with porous solid
\vspace*{1mm}
\item[-] Full saturation and partial, unsaturated soils
\vspace*{1mm}
\item[-] Statics and dynamics of coupled systems
%
\end{itemize}
\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}
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\begin{frame}
\frametitle{FEM Discretization, Stress-Strain}
%\begin{itemize}
%%
%
%\vspace*{1mm}
% \item[-] Stress-strain relation:
%\\
$
\Delta \hat{\sigma}_{ab}
=
E^{EP}_{abcd}
\left( \Delta \hat{\epsilon}_{cd} - \Delta \epsilon_{cd}^{0} \right)
+
\Delta \sigma_{ab}^{0}
$
%\end{itemize}
%\vspace*{-5mm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{FEM Discretization, Stiffness Tensor Symmetries}
\begin{itemize}
\item[-] Minor symmetry of stiffness tensor:
$^{EP}E_{abcd} = E^{EP}_{bacd}= E^{EP}_{abdc}$
\vspace*{5mm}
\item[-] Major symmetry is not necessary, non-associated plasticity
$^{EP}E_{abcd} \ne E^{EP}_{cdab}$
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
% \frametitle{FEM Discretization, Equation}
%
%
% \begin{itemize}
%
%
%
% \vspace*{2mm}
% \item[-] Virtual displacements cannot be zero, they cancel out
%
% \vspace*{2mm}
% \item[-] Final FEM equations:
% $
% 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)}
%
%
%
% \end{itemize}
%
%
% \end{frame}
%
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%
% \begin{frame}
% \frametitle{FEM Discretization, Tensors}
%
%
% % \begin{itemize}
%
%
% \vspace*{2mm}
% \item[-] Final FEM equations:
% $
% 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*{3mm}
% \item[-] Mass matrix/tensor:
% $
% M_{IacJ}
% =
% \int_{V^{m} } H_{J} \; \delta_{ac} \; \rho \; H_{I} \; dV^{m}
% $
%
%
% \vspace*{3mm}
% \item[-] Stiffness matrix/tensor:
% $
% K_{IacJ}
% =
% \int_{V^{m} } H_{I,b} \; E_{abcd} \; H_{J,d} \; dV^{m}
% $
%
% \vspace*{3mm}
% \item[-] Load vector/tensor:
% $
% F_{Ia}
% =
% \int_{V^{m} } f^{B}_{a} \; H_{I} \; dV^{m}
% +
% \int_{S^{m} } f^{S}_{a} \; H_{I} \; dS^{m}
% +
% \int_{V^{m} } E_{abcd} \; \epsilon_{cd}^{0} \; H_{I,b} \; dV^{m}
% +
% \int_{V^{m} } \sigma_{ab}^{0} \; H_{I,b} \; dV^{m}
% $
%
%
%
%
% \end{itemize}
%
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
% \frametitle{FEM Discretization, Matrix Form}
%
%
% \begin{itemize}
%
%
%
% \vspace*{4mm}
% \item[-] FEM matrix form:
% $
% M_{PQ} \; \ddot{\bar{u}}_{P}
% +
% K_{PQ} \; \bar{u}_{P}
% =
% F_{Q}
% $
%
%
%
%
%
% \end{itemize}
%
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
% \frametitle{FEM, Matrix Form, Residuals, Equilibrium}
%
%
% \begin{itemize}
%
%
% \vspace*{4mm}
% \item[-] Nonlinear FEM residuals, equilibrium:
%
% $
% R_{Q} =
% F_{Q} -
% \left(
% M_{PQ} \; \ddot{\bar{u}}_{P}
% +
% K_{PQ} \; \bar{u}_{P}
% \right)
% $
%
%
% \vspace*{-2.5cm}
% \begin{figure}[!hbpt]
% \begin{flushright}
% \includegraphics[width=3cm]{/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{flushright}
% \vspace*{-0.5cm}
% \end{figure}
% %
%
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% \end{itemize}
%
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% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Notation}
%$\sigma_{ij}^{''} = \sigma_{ij} - \alpha \delta_{ij} p $
%Notation
\begin{itemize}
\item[-] Effective stress $\sigma_{ij}^{''} = \sigma_{ij} - \alpha \delta_{ij} p $
\item[-] $\sigma_{ij}$ - the total Cauchy stress in the mixture
\item[-] $\alpha=1-K_T/K_S \approx 1$
\item[-] $p$ - the pore fluid pressure
\item[-] $u_i$ - the displacement of the solid skeleton
\item[-] $w_i$ - disp. of fluid phase relative to the skeleton of solid
\item[-] $\varepsilon_{ij} =\frac{1}{2}(u_{i,j}+u_{j,i})$ -
strain
increment of the solid phase,
%\item[] $\omega_{ij} =\frac{1}{2}(du_{i,j}-du_{j,i})$, the rotation increment
%of the solid phase,
\item[-] $\rho,\rho_s,\rho_f$ - densities of mixture, solid phase and water
\item[-] $n=V_{voids}/V_{total}$ - porosity,
\item[-] $\dot{w}_{i,i}$ - rate of change of volume of fluid / volume of mixture
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Governing Equations}
\begin{itemize}
\item[-] Equilibrium Equation of the Mixture:
$
\sigma_{ij,j}-\rho \ddot{u}_i-\rho_f[\ddot{w}_i+
\underline{\dot{w}_j\dot{w}_{i,j}}]+\rho b_i=0
$
$
\rho=\frac{M_t}{V_t}
=\frac{M_s+M_f}{V_t}
=\frac{V_s\rho_s + V_f\rho_f}{V_t}
=\frac{V_f}{V_t}\rho_f+\frac{V_t-V_f}{V_t}\rho_s
= n\rho_f+(1-n)\rho_s
$
\vspace*{1mm}
\item[-] Equilibrium Equation of the Fluid
$
-p_{,i}- R_i - \rho_f \ddot{u}_i-\rho_f[\ddot{w}_i+
\underline{\dot{w}_j \dot{w}_{i,j}}]/n+\rho_f b_i=0
$
%
$R_i=k_{ij}^{-1} \dot{w}_j \qquad \textrm{or} \qquad
R_i=k^{-1} \dot{w}_i$
\vspace*{1mm}
\item[-] Flow Conservation Equation
$ %
\dot{w}_{i,i}+\alpha \dot{\varepsilon}_{ii}+\frac{\dot{p}}{Q}+\underline{n
\frac{\dot{\rho_f}}{\rho_f}+\dot{s}_0}=0
$
%
$\frac{1}{Q}\equiv\frac{n}{K_f}+\frac{\alpha-n}{K_s}\cong \frac{n}{K_f}+\frac{1-n}{K_s}$
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Simplified Governing Equations}
\begin{itemize}
\item[-] Equilibrium Equation of the Mixture:
%
\begin{eqnarray}
\sigma_{ij,j}-\rho \ddot{u}_i-\rho_f \ddot{w}_i+\rho b_i=0
\nonumber
\end{eqnarray}
\vspace*{3mm}
\item[-] Equilibrium Equation of the Fluid
%
\begin{eqnarray}
- p_{,i}- R_i -\rho_f \ddot{u}_i-\frac{\rho_f \ddot{w}_i}{n}+\rho_f b_i=0
\nonumber
\end{eqnarray}
\vspace*{3mm}
\item[-] Flow Conservation Equation
%
\begin{eqnarray}
\dot{w}_{i,i}+\alpha \dot{\varepsilon}_{ii}+\frac{\dot{p}}{Q}=0
\nonumber
\end{eqnarray}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Modified Governing Equations}
\vspace*{5mm}
$
U_i=u_i+\frac{w_i}{n}
$
\hspace*{10mm}
\includegraphics[width=2cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Theory_FEM_Formulation/tex_works_LectureNotes_Figures_Darcy_vs_Real_01.pdf}
%
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Theory_FEM_Formulation/tex_works_LectureNotes_Figures_Darcy_vs_Real_01.pdf}
% \end{center}
% \end{figure}
\begin{eqnarray}
\sigma_{ij,j}^{''}-(\alpha-n) p_{,i}+(1-n) \rho_s b_i-(1-n) \rho_s \ddot{u}_i +
n R_i=0
\nonumber
\end{eqnarray}
%%
%
\begin{eqnarray}
-n p_{,i}+n \rho_f b_i-n \rho_f \ddot{U}_i
- n R_i=0
\nonumber
\end{eqnarray}
%
\begin{eqnarray}
-n \dot{U}_{i,i}=(\alpha-n) \dot{\varepsilon}_{ii}+\frac{1}{Q} \dot{p}
\nonumber
\end{eqnarray}
\begin{itemize}
\item[]
$u_i$: three solid displacement
\item[]
$p$: pore pressure
\item[]
$U_i$: three fluid displacement
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{FEM u-p-U Discretization}
%Detailed derivation in Lecture Notes, Part I, Chapter 102
{\small
\begin{eqnarray}
\left[ \begin{array}{ccc}
M_s & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & M_f
\end{array} \right]
\left[ \begin{array}{c}
\ddot{\overline{u}} \\
\ddot{\overline{p}} \\
\ddot{\overline{U}}
\end{array} \right]
+
\left[ \begin{array}{ccc}
C_1 & 0 & -C_2 \\
0 & 0 & 0 \\
-C_2^T & 0 & C_3 \\
\end{array} \right]
\left[ \begin{array}{c}
\dot{\overline{u}} \\
\dot{\overline{p}} \\
\dot{\overline{U}}
\end{array} \right]
+ \nonumber \\
\left[ \begin{array}{ccc}
K^{EP} & -G_1 & 0 \\
-G_1^T & -P & -G_2^T \\
0 & -G_2 & 0
\end{array} \right]
\left[ \begin{array}{c}
\overline{u} \\
\overline{p} \\
\overline{U}
\end{array} \right] \nonumber
=
\left[ \begin{array}{c}
\overline{f}_s \\
0 \\
\overline{f}_f
\end{array} \right]
\nonumber
\end{eqnarray}
}
%
%
%
%
%\vspace*{-5mm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{FEM u-p-U Discretization, Index Form}
%Detailed derivation in Lecture Notes, Part I, Chapter 102
{\footnotesize
\begin{eqnarray}
\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]
\end{eqnarray}
}
%
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{FEM u-p-U Matrices/Tensors}
{\footnotesize
\begin{eqnarray}
\hspace*{-8mm}
\boldsymbol{M_s}=(M_s)_{KijL}=\int_{\Omega} H_K^u (1-n) \rho_s \delta_{ij} H_L^u d\Omega
~~ \mbox{;} ~~
\boldsymbol{M_f}=(M_f)_{KijL}=\int_{\Omega} H_K^U n \rho_f \delta_{ij} H_L^U d\Omega \nonumber\\
%
\boldsymbol{C_1}=(C_1)_{KijL}=\int_{\Omega} H_K^u n^2 k_{ij}^{-1} H_L^u d\Omega
~~ \mbox{;} ~~
\boldsymbol{C_2}=(C_2)_{KijL}=\int_{\Omega} H_K^u n^2 k_{ij}^{-1} H_L^U d\Omega \nonumber\\
\boldsymbol{C_3}=(C_3)_{KijL}=\int_{\Omega} H_K^U n^2 k_{ij}^{-1} H_L^U d\Omega \nonumber\\
%
\boldsymbol{K^{EP}}=(K^{EP})_{KijL}=\int_{\Omega} H_{K,m}^u D_{imjn} H_{L,n}^u d\Omega \nonumber\\
\boldsymbol{G_1}=(G_1)_{KiM}=\int_{\Omega} H_{K,i}^u (\alpha-n) H_M^p d\Omega
~~ \mbox{;} ~~
\boldsymbol{G_2}=(G_2)_{KiM}=\int_{\Omega} n H_{K,i}^U H_M^p d\Omega \nonumber\\
\boldsymbol{P}=P_{NM}=\int_{\Omega} H_N^p \frac{1}{Q} H_M^p d\Omega \nonumber
\end{eqnarray}
}
%
%
%
%
%\vspace*{-5mm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{FEM u-p-U Loads}
%\newpage
{\footnotesize
\hspace*{-8mm}
\begin{eqnarray}
\overline{f}_{Ki}^{solid}&=&(f_1^u)_{Ki}-(f_4^u)_{Ki}+(f_5^u)_{Ki} \nonumber\\
\overline{f}_{Ki}^{fluid}&=&-(f_1^U)_{Ki}+(f_2^U)_{Ki} \nonumber\\
(f_1^u)_{Ki}&=&\int_{\Gamma_t} H_K^u n_j \sigma_{ij}^{''} d\Gamma \nonumber\\
(f_4^u)_{Ki}&=&\int_{\Gamma_p} H_K^u (\alpha-n) n_i p d\Gamma \nonumber\\
(f_5^u)_{Ki}&=&\int_{\Omega} H_K^u (1-n) \rho_s b_i d\Omega \nonumber\\
(f_1^U)_{Ki}&=&\int_{\Gamma_p} n H_K^U n_i p d\Gamma \nonumber\\
(f_2^U)_{Ki}&=&\int_{\Omega} n H_K^U \rho_f b_i d\Omega \nonumber
\label{eqn_69_u-p-U}
\end{eqnarray}
}
%
%\vspace*{-5mm}
\end{frame}
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\begin{frame}
\frametitle{Coupled FEM Discretization, Matrix Form}
\begin{itemize}
\vspace*{1mm}
\item[-] Coupled FEM, full matrix form:
\begin{eqnarray*}
\left[ \begin{array}{ccc}
M_s & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & M_f
\end{array} \right]
\left[ \begin{array}{c}
\ddot{\overline{u}} \\
\ddot{\overline{p}} \\
\ddot{\overline{U}}
\end{array} \right]
+
\left[ \begin{array}{ccc}
C_1 & 0 & -C_2 \\
0 & 0 & 0 \\
-C_2^T & 0 & C_3 \\
\end{array} \right]
\left[ \begin{array}{c}
\dot{\overline{u}} \\
\dot{\overline{p}} \\
\dot{\overline{U}}
\end{array} \right]
+
\\
+
\left[ \begin{array}{ccc}
K^{EP} & -G_1 & 0 \\
-G_1^T & -P & -G_2^T \\
0 & -G_2 & 0
\end{array} \right]
\left[ \begin{array}{c}
\overline{u} \\
\overline{p} \\
\overline{U}
\end{array} \right]
=
\left[ \begin{array}{c}
\overline{f}_s \\
0 \\
\overline{f}_f
\end{array} \right]
\nonumber
\label{eqn_67_u-p-U}
\end{eqnarray*}
%
\vspace*{3mm}
\item[-] Coupled FEM, generalized matrix form:
$
M_{PQ} \; \ddot{\bar{u}}_{P}
+
C_{PQ} \; \dot{\bar{u}}_{P}
+
K_{PQ} \; \bar{u}_{P}
=
F_{Q}
$
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Coupled FEM, Residuals, Equilibrium}
\begin{itemize}
\vspace*{6mm}
\item[-] Nonlinear coupled FEM residuals, \\
equilibrium:
%
%\begin{flushleft}
%\begin{eqnarray*}
$
R_{Q} =
$
\\
$
F_{Q} -
\left(
M_{PQ} \; \ddot{\bar{u}}_{P}
+
C_{PQ} \; \dot{\bar{u}}_{P}
+
K_{PQ} \; \bar{u}_{P}
\right)
$
%\nonumber
%\end{eqnarray*}
%\end{flushleft}
\vspace*{-3.5cm}
\begin{figure}[!hbpt]
\begin{flushright}
\includegraphics[width=3cm]{/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{flushright}
\vspace*{-0.5cm}
\end{figure}
%
\end{itemize}
\end{frame}
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\subsection{Analysis Examples}
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\begin{frame}
\frametitle{Coupled Analysis in Practice}
\begin{itemize}
\vspace*{2mm}
\item[-] The coupled u-p-U formulation is very powerful
\vspace*{4mm}
\item[-] The u-p-U formulation is useful for many problems
\vspace*{4mm}
\item[-] ESSI: internal fluids problems
%, porous solids -- pore fluid
\vspace*{4mm}
\item[-] ESSI: external fluid problems
%, porous solids -- pore fluid -- external fluid
% \vspace*{2mm}
% \item[-]
%
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% \vspace*{2mm}
% \item[-]
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% \item[-]
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\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Liquefaction as Base Isolation}
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=4.8truecm]{/home/jeremic/tex/works/Conferences/2017/Slovenia_IAEA_short_course/present/SSI-Site_Response_Analysis/Liquefaction_04.jpg}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Liquefaction, Wave Propagation}
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=6.0truecm]{/home/jeremic/tex/works/Conferences/2017/Slovenia_IAEA_short_course/present/SSI-Site_Response_Analysis/Liquefaction_01.jpg}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Liquefaction, Stress-Strain Response}
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=5.5truecm]{/home/jeremic/tex/works/Conferences/2017/Slovenia_IAEA_short_course/present/SSI-Site_Response_Analysis/Liquefaction_03.jpg}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Pile in Liquefiable Soil}
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=2.5truecm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_Cyclic_Mobility_and_Liquefaction/tex_works_Papers_2008_Pile_in_liquefied_soil_upU_final_FEmesh3D.pdf}
\hfill
\includegraphics[width=3.2truecm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_Cyclic_Mobility_and_Liquefaction/tex_works_Papers_2008_Pile_in_liquefied_soil_upU_final_FEmeshPileBeam.pdf}
%\hfill
\includegraphics[width=4.7truecm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_Cyclic_Mobility_and_Liquefaction/tex_works_Papers_2008_Pile_in_liquefied_soil_upU_final_SFSIModelSetup_upU_01.pdf}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Pile in Liquefiable Soil}
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=6.7truecm]{/home/jeremic/tex/works/Conferences/2018/Oersted-DONG-Energy/present/Pile_in_liquefied_soil.jpg}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Buoyant Force Effects}
\begin{figure}[!H]
\hspace*{-10mm}
\includegraphics[width=5.5cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/upU_element_type_annotation.pdf}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/bouyant_displacement.pdf}
\hspace*{-10mm}
\end{figure}
% - %
% - % \begin{tikzpicture}[remember picture,overlay]
% - % \node[anchor=south west,inner sep=0pt] at ($(current page.south west)+(7.5cm,2.5cm)$) {
% - % \includegraphics[width=0.4\textwidth]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/bouyant_displacement.pdf}};
% - % \end{tikzpicture}
% -
% - % \vspace{-1.4cm}
% \begin{itemize}
% \item \scriptsize Upward structural displacement under buoyant force
% \end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Building on Liquefiable Soil}
\vspace*{16mm}
\noindent
Plastic Strain \hspace*{40mm} Pore Fluid Pressures
\vspace*{-1mm}
% local
%\vspace*{-2mm}
\begin{center}
\hspace*{-20mm}
\movie[label=show3,width=5.5cm,showcontrols]
%\movie[label=show3,width=61mm, showcontrols]
{\includegraphics[width=59mm]
{/home/jeremic/tex/works/Thesis/HanYang/Liquefaction_models_Nov-Dec2021/Plastic_Strain.jpg}}
{/home/jeremic/tex/works/Thesis/HanYang/Liquefaction_models_Nov-Dec2021/Plastic_Strain.mp4}
%\hspace*{-2mm}
%\hfill
\movie[label=show3,width=5.6cm,showcontrols]
%\movie[label=show3,width=61mm, showcontrols]
{\includegraphics[width=60mm]
{/home/jeremic/tex/works/Thesis/HanYang/Liquefaction_models_Nov-Dec2021/Pore_Presure.jpg}}
{/home/jeremic/tex/works/Thesis/HanYang/Liquefaction_models_Nov-Dec2021/Pore_Presure.mp4}
\hspace*{-16mm}
\end{center}
% local
\vspace*{-1mm}
% online
\begin{center}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/building_on_liquefiable_soil_ETH_model/Plastic_Strain.mp4}
{\tiny (MP4)}
%
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/building_on_liquefiable_soil_ETH_model/Pore_Presure.mp4}
{\tiny (MP4)}
\end{center}
% online
\end{frame}
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\begin{frame}
\frametitle{Solid/Structure-Fluid Interaction}
% local
% local
% local %\vspace*{-5mm}
% local \begin{center}
% local % \hspace*{-15mm}
% local \movie[label=show3,width=9cm,poster,autostart,showcontrols]
% local {\includegraphics[width=8.5cm]
% local {/home/jeremic/tex/works/Conferences/2017/DOE_Project_Review_Meeting_LBNL_09June2017/Present/Solid-Fluid-Interaction.jpg}}
% local {Solid_Fluid_Interaction.mp4}
% local \end{center}
% local
% local
% local
% online
% online
% online
% online \begin{center}
% online %\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Fluid_Solid_interaction/Solid_Fluid_Interaction.mp4}
% online \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Fluid_Solid_interaction/Solid_Fluid_Interaction_NEW.mpeg}
% online {\includegraphics[width=90mm]{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Fluid_Solid_interaction/Solid_Fluid_Interaction.jpg}}
% online \end{center}
% online % online
% online
% online
%\vspace*{-5mm}
\begin{center}
% \hspace*{-15mm}
%\movie[label=show3,width=9cm,poster,autostart,showcontrols]
\movie[label=show3,width=9cm,showcontrols]
{\includegraphics[width=8.5cm]
{/home/jeremic/tex/works/Conferences/2017/DOE_Project_Review_Meeting_LBNL_09June2017/Present/Solid-Fluid-Interaction.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Fluid_Solid_interaction/Solid_Fluid_Interaction_NEW.mpeg}
\end{center}
\begin{flushleft}
\vspace*{-15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Fluid_Solid_interaction/Solid_Fluid_Interaction.mp4}
% \href{./homo_50m-mesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
\end{flushleft}
%
\end{frame}
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%
%
%
% \subsection{Nonlinear FEM Analysis Process}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
% \frametitle{Nonlinear FEM Analysis Process}
%
%
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=3.5cm]{/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}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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{\ }
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Nonlinear FEM}
%
% \begin{itemize}
%
% \vspace*{-8mm}
% \item[-] Nonlinear FEM, global level, iterations
%
% \vspace*{1mm}
% \item[-] Nonlinear FEM, constitutive level, iterations
%
%
% \vspace*{1mm}
% \item[-] No equilibrium check, explicit
%
% \vspace*{1mm}
% \item[-] Equilibrium check, implicit
%
% \vspace*{1mm}
% \item[-] Explicit, no equilibrium on constitutive and \\
% global levels
%
% \vspace*{1mm}
% \item[-] Implicit, equilibrium on constitutive and \\
% global levels
%
%
%
% \vspace*{1mm}
% \item[-] Gradually building confidence in inelastic modeling
%
%
% \end{itemize}
%
% \vspace*{-5.5cm}
% \begin{figure}[!hbpt]
% \begin{flushright}
% \includegraphics[width=2.0cm]{/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{flushright}
% \vspace*{-0.5cm}
% \end{figure}
% %
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% \end{frame}
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%
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%
% \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}
% \\
% \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 = 4cm]{/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}
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\begin{frame}
\frametitle{Analysis of ESSI Systems }
\begin{itemize}
\vspace*{1mm}
\item[-] Soil, elastic, elastic-plastic/inelastic
\begin{itemize}
\item[-] Dry, single phase
\item[-] Unsaturated/partially saturated
\item[-] Fully saturated
\end{itemize}
\vspace*{1mm}
\item[-] Contact/Interface/Joint, inelastic
\begin{itemize}
\item[-] Dry, single phase, Normal, hard and soft, gap open/close
\item[-] Dry, single phase, Tangential, friction, nonlinear
\item[-] Fully saturated, suction/excess pressure, buoyant force
\end{itemize}
\vspace*{1mm}
\item[-] Structure inelasticity/damage
\begin{itemize}
\item[-] Nonlinear/inelastic 1D fiber beam
\item[-] Nonlinear/inelastic 3D shell/wall/plate element
\end{itemize}
\vspace*{1mm}
\item[-] Fluid-Solid interaction, open surface fluid
\end{itemize}
%
\end{frame}
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% \subsection{Solid Elastic-Plastic Examples}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Contact Inelastic Examples}
%
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\begin{frame}
\frametitle{Summary}
%OVDE
\begin{itemize}
\item[-] Reduction of modeling uncertainty
\vspace*{4mm}
\item[-] Choice of analysis level of sophistication
\vspace*{4mm}
\item[-] Numerical analysis to predict and inform
% \vspace*{1mm}
% \item[-] Change of demand due to inelastic effects
%
% \begin{itemize}
% \item[-] Reduction of dynamic motions
% \item[-] Increase in deformations
% \end{itemize}
%
\vspace*{4mm}
\item[-] Engineer needs to know!
%
%\vspace*{4mm}
% \item[-] Sophisticated analysiscarefully and in phases
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% \vspace*{4mm}
% \item[-] Gradually building confidence in inelastic modeling
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% \item The Real ESSI Simulator system used for all the model
% development, modeling/simulations and post-processing
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\vspace*{4mm}
\item[-] The Real ESSI Simulator system
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\hspace*{30mm} \href{http://real-essi.info}{http://real-essi.info}
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% %\vspace*{1mm}
% \item Nonlinear/Inelastic Earthquake Soil Structure Interaction (ESSI)
% short course offered this fall in San Francisco, more info at
% \href{http://real-essi.info}{http://real-essi.info}
\end{itemize}
\end{frame}
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\end{document}