\documentclass{beamer}
%\documentclass[handout]{beamer}
%\usepackage{pgfpages}
%\pgfpagelayout{2 on 1}[letterpaper, border shrink=5mm]
\mode{\setbeamercolor{background canvas}{bg=black!3}}
% %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/2005January/000780.html
%
% \ifx\pdfoutput\undefined
% % we are running LaTeX, not pdflatex
% \usepackage{graphicx}
% \else
% % we are running pdflatex, so convert .eps files to .pdf
% \usepackage[pdftex]{graphicx}
% \usepackage{epstopdf}
% \fi
%*****************************************
% % for not showing eq numbers unless eq is references
% \usepackage[fleqn,tbtags]{mathtools}
% \mathtoolsset{
% showonlyrefs
% }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ovo je za cirilicu
\input vuk.def
%\newfont{\cyr}{wncyr10 scaled 1200}
%\newfont{\cyrimena}{wncyb10 scaled 1800}
%\newfont{\cyrnaslov}{wncyb10 scaled 2600}
%\newfont{\cyrpodnaslov}{wncyr10 scaled 1400}
\newfont{\cyr}{wncyss10 scaled 1200}
\newfont{\cyrimena}{wncyss10 scaled 2000}
\newfont{\cyrnaslov}{wncyss10 scaled 2800}
\newfont{\cyrpodnaslov}{wncyss10 scaled 1400}
\newfont{\cyn}{wncyss10 scaled 2200}
%% JB sig
\newfont{\cyrjb}{wncyr10 scaled 600}
\newcommand{\JB}
{{\cyrjb \lower0.50ex\hbox{\uppercase{J}}\kern.58em\hbox{\uppercase{B}}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setbeamertemplate{navigation symbols}{}
\mode
{
% \usetheme{Marburg} % ima naslov i sadrzaj sa desne strane
% \usetheme{Hannover} % ima naslov i sadrzaj sa leve strane
% \usetheme{Singapore} % ima sadrzaj i tackice gore
% \usetheme{Antibes} % ima sadrzaj gore i kao graf ...
% \usetheme{Berkeley} % ima sadrzaj desno
% \usetheme{Berlin} % ima sadrzaj gore i tackice
% \usetheme{Goettingen} % ima sadrzxaj za desne strane
% \usetheme{Montpellier} % ima graf sadrzaj gore
% \usetheme{Warsaw}
% \usetheme{Warsaw}
\usetheme{Dresden}
\usecolortheme[RGB={20,0,128}]{structure}
% or ...
\setbeamercovered{transparent}
% \setbeamercovered{transparent}
% or whatever (possibly just delete it)
% \usecolortheme{albatross} % teget sa svetlim slovima
% \usecolortheme{beetle} % siva pozadina (vrh plav)
% \usecolortheme{seagull} % sivo
%%%%%%%
% \usecolortheme{BorisJeremic}
%%%%%%%
% \usecolortheme{rose}
% \usefonttheme[onlylarge]{structuresmallcapsserif}
% \usefonttheme{structuresmallcapsserif}
}
\definecolor{mycolor}{rgb}{0,0.08,0.45}%
\usepackage[english]{babel}
\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\newcommand{\ud}{{\rm d}}
\usepackage{array}
%%%% HYPERREF HYPERREF HYPERREF HYPERREF HYPERREF
%%%% HYPERREF HYPERREF HYPERREF HYPERREF HYPERREF
\definecolor{webgreen}{rgb}{0, 0.15, 0} % less intense green
\definecolor{webblue}{rgb}{0, 0, 0.15} % less intense blue
\definecolor{webred}{rgb}{0.15, 0, 0} % less intense red
%\usepackage[colorlinks=true,linkcolor=webblue,citecolor=webred,urlcolor=webgreen]{hyperref}
\usepackage{hyperref}
\hypersetup{
pdfmenubar=true,
pdftoolbar=true,
pdfpagemode={None}
}
\usepackage{pause}
% or whatever
%\usepackage{html}
%\usepackage{url}
\usepackage[latin1]{inputenc}
% or whatever
\usepackage{times}
\usepackage[T1]{fontenc}
% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.
%\title{NRC Staff Capacity Building: \\
% Micromechanical Origins of ElastoPlasticity }
\title{Earthquake Soil Structure Interaction for \\
Nuclear Power Plants, \\
Modeling and Computational Issues}
%\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]{universitylogo}{/home/jeremic/BG/amblemi/ucdavis_logo_blue_sm}
\pgfdeclareimage[height=0.7cm]{lbnllogo}{/home/jeremic/BG/amblemi/lbnllogo}
\author[Jeremi{\'c} et al.] % (optional, use only with lots of authors)
{B.~Jeremi{\'c} \\
N.~Tafazzoli, P.~Tasiopoulou, J.A.~Abell~Mena,
B.~Kamrani, C.G.~Jeong,
F.~Pisan{\`o}, M.~Martinelli,
K.~Sett, M.~Taiebat
}
%  Give the names in the same order as the appear in the paper.
%  Use the \inst{?} command only if the authors have different
% affiliation.
%\institute[Computational Geomechanics Group \hspace*{0.3truecm}
\institute[\pgfuseimage{universitylogo}\hspace*{0.1truecm}\pgfuseimage{lbnllogo}] % (optional, but mostly needed)
{ Professor, University of California, Davis\\
% and\\
Faculty Scientist, Lawrence Berkeley National Laboratory, Berkeley }
%  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 CompDyn, Kos, Greece, June, 2013}
\subject{}
% This is only inserted into the PDF information catalog. Can be left
% out.
% If you have a file called "universitylogofilename.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]{universitylogo}{/home/jeremic/BG/amblemi/ucdavis_logo_gold_lrg}
%\logo{\pgfuseimage{universitylogo}}
% \pgfdeclareimage[height=0.5cm]{universitylogo}{universitylogofilename}
% \logo{\pgfuseimage{universitylogo}}
% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
% \AtBeginSection[]
\AtBeginSubsection[]
{
\begin{frame}
\frametitle{Outline}
\tableofcontents[currentsection,currentsubsection]
% \tableofcontents[currentsection]
\end{frame}
}
% If you wish to uncover everything in a stepwise fashion, uncomment
% the following command:
\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\titlepage
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Outline}
\tableofcontents
% You might wish to add the option [pausesections]
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Structuring a talk is a difficult task and the following structure
% may not be suitable. Here are some rules that apply for this
% solution:
%  Exactly two or three sections (other than the summary).
%  At *most* three subsections per section.
%  Talk about 30s to 2min per frame. So there should be between about
% 15 and 30 frames, all told.
%  A conference audience is likely to know very little of what you
% are going to talk about. So *simplify*!
%  In a 20min talk, getting the main ideas across is hard
% enough. Leave out details, even if it means being less precise than
% you think necessary.
%  If you omit details that are vital to the proof/implementation,
% just say so once. Everybody will be happy with that.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Motivation}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{}
%
% \begin{itemize}
% %\vspace*{0.3cm}
% \item
%
% \item
% \end{itemize}
% \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Problem  Solution}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{The Problem}
\begin{itemize}
\item Seismic response of Nuclear Power Plants ($f \le 50$Hz! ($20$Hz))
\vspace*{0.1cm}
\item 3D, inclined seismic motions: body and surface waves
\vspace*{0.1cm}
\item Inelasticity (elastic, damage, plastic behavior of materials: soil,
rock,
concrete, steel, rubber, contact, etc.)
\vspace*{0.1cm}
\item Full coupling of pore fluids with soil, rock and
concrete skeleton, including buoyancy effects
\vspace*{0.1cm}
\item Uncertainty in seismic sources, path, soil/rock
and structural response
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Solution}
\begin{itemize}
%\vspace*{0.3cm}
\item {\bf Physics based modeling and simulation} of seismic behavior of
soilstructure systems (NPP structures, components and systems)
\vspace*{0.1cm}
\item Development and use of {\bf high fidelity} time domain,
nonlinear numerical models,
in {\bf deterministic} and {\bf probabilistic} spaces, for
licensing and professional practice (every day use)
\vspace*{0.1cm}
\item Accurate following of the {\bf flow of seismic
energy} (input and dissipation) within soilstructure NPP system
\vspace*{0.1cm}
\item {\bf Directing}, in space and time, with {\bf high (known)
confidence}, seismic energy flow in the soilfoundationstructure system
%\vspace*{0.1cm}
% \item {\bf Education} for researchers, professional practice.
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{NPP Model(s)}
\vspace*{0.5cm}
\begin{figure}[!hbpt]
\begin{center}
%\hspace*{0.5cm}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/03_cropped.jpg}
%\hspace*{0.5cm}
%\vspace*{0.5cm}
\includegraphics[width=3cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/02_cropped.jpg}
%\hspace*{0.5cm}
%\vspace*{1.5cm}
\includegraphics[width=1.5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/04_cropped.jpg}
%\hspace*{0.5cm}
\end{center}
\end{figure}
%
\end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \section{Challenges}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Uncertainty in Modeling}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Modeling Uncertainty}
\begin{itemize}
%\vspace*{0.3cm}
\item Simplified (or inadequate/wrong) modeling: important features are
missed (seismic ground motions, etc.)
\vspace*{0.2cm}
\item Introduction of uncertainty and (unknown) lack of accuracy in results due
to use of unverified simulation tools (software quality, etc.)
\vspace*{0.2cm}
\item Introduction of uncertainty and (unknown) lack of accuracy in results due
to use of unvalidated models (due to lack of quality validation experiments)
% (still missing data, experiments under
% uncertainty, for more see below)
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Complexity of and Uncertainty in Ground Motions}
\begin{itemize}
%\vspace*{0.3cm}
\item 6D (3 translations, 3 rotations)
\vspace*{0.3cm}
\item Vertical motions usually neglected
\vspace*{0.3cm}
\item Rotational components usually not measured and neglected
\vspace*{0.3cm}
\item Lack of models for such 6D motions (from measured data))
\vspace*{0.3cm}
\item Sources of uncertainties in ground motions (Source, Path (Rock), Soil/Rock))
\end{itemize}
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Uncertainty in Modeling Material}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Material Behavior Inherently Uncertain}
%\begin{itemize}
%\vspace*{0.5cm}
%\item
%Material behavior is inherently uncertain (concrete, metals, soil, rock,
%bone, foam, powder etc.)
\begin{itemize}
\vspace*{0.5cm}
\item (a) Spatial \\
variability
\vspace*{0.5cm}
\item Pointwise \\
uncertainty, \\
(b) testing \\
error, \\
(c) transformation \\
error
\end{itemize}
% \vspace*{0.5cm}
% \item Failure mechanisms related to spatial variability (strain localization and
% bifurcation of response)
%
% \vspace*{0.5cm}
% \item Inverse problems
%
% \begin{itemize}
%
% \item New material design, ({\it pointwise})
%
% \item Solid and/or structure design (or retrofits), ({\it spatial})
%
% \end{itemize}
%\end{itemize}
\vspace*{5cm}
\begin{figure}[!hbpt]
%\nonumber
%\begin{center}
\begin{flushright}
%\includegraphics[height=5.0cm]{/home/jeremic/tex/works/Conferences/2006/KragujevacSEECCM06/Presentation/MGMuzorak01.jpg}
\includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/FrictionAngleProfile.jpg}
\\
\mbox{(Mayne et al. (2000) }
\end{flushright}
%\end{center}
%\end{center}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Soil Uncertainties and Quantification}
%
% \begin{itemize}
% %
% %\vspace*{0.5cm}
% \item Natural variability of soil deposit (Fenton 1999)
%
% \begin{itemize}
%
% \item Function of soil formation process
%
% \end{itemize}
%
%
% %
% \vspace*{0.2cm}
% \item Testing error (Stokoe et al. 2004)
%
% \begin{itemize}
%
% \item Imperfection of instruments
%
% \item Error in methods to register quantities
%
% \end{itemize}
%
% %
% \vspace*{0.2cm}
% \item Transformation error (Phoon and Kulhawy 1999)
%
% \begin{itemize}
%
% \item Correlation by empirical data fitting (e.g. CPT data $\rightarrow$ friction angle etc.)
%
% \end{itemize}
%
% \end{itemize}
%
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
%
%
% \frametitle{SPT Based Determination of Shear Strength}
%
%
%
%
% \begin{figure}[!hbpt]
% \begin{center}
% %
% \includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/ShearStrength_RawData_and_MeanTrendMod.pdf}
% \hfill
% \includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/ShearStrength_Histogram_PearsonIVFineTunedMod.pdf}
% %
% \end{center}
% \end{figure}
%
% \vspace*{0.3cm}
% Transformation of SPT $N$value $\rightarrow$ undrained shear
% strength, $s_u$ (cf. Phoon and Kulhawy (1999B)
%
% Histogram of the residual
% (w.r.t the deterministic transformation
% equation) undrained strength,
% along with fitted probability density function
% (Pearson IV)
% \end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
%
%
% \frametitle{SPT Based Determination of Young's Modulus}
%
%
% \begin{figure}[!hbpt]
% \begin{center}
% %
% \includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01Ed.pdf}
% \hfill
% \includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_Histogram_Normal_01Ed.pdf}
% %
% \end{center}
% \end{figure}
%
% \vspace*{0.3cm}
% Transformation of SPT $N$value $\rightarrow$ 1D Young's modulus, $E$ (cf. Phoon and Kulhawy (1999B))
%
% Histogram of the residual (w.r.t the deterministic transformation equation) Young's modulus, along with fitted probability density function
%
% \end{frame}
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Errors in Scientific Software}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Errors in Scientific Software: The T Experiments}
%
%
% \begin{itemize}
%
% % \vspace*{0.2truecm}
% \item Les Hatton, Kingston University (formerly of Oakwood Comp. Assoc.)
%
% \vspace*{0.1truecm}
% \item "Extensive tests showed that many software codes widely used in science
% and engineering are not as accurate as we would like to think."
%
% \vspace*{0.1truecm}
% \item "Better software engineering practices would help solve this problem,"
%
% \vspace*{0.1truecm}
% \item "Realizing that the problem exists is an important first step."
%
%
% \vspace*{0.1truecm}
% \item Large experiment over 4 years measuring faults (T1) and failures (T2)
% of scientific and engineering codes
%
%
%
%
%
% \end{itemize}
%
%
%
%
%
% \end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{The T2 Experiments}
%
%
%
%
%
%
% \begin{itemize}
%
% \item Specific application area: seismic data processing (inverse analysis)
%
% \vspace*{0.2truecm}
% \item Echo sounding of underground and reconstructing "images" of
% subsurface geological structure
%
% \vspace*{0.2truecm}
% \item Nine mature packages, using {\bf same algorithms}, on a {\bf same data set}!
%
% \vspace*{0.2truecm}
% \item 14 primary calibration points for results check
%
% \vspace*{0.2truecm}
% \item Results "fascinating and disturbing"
%
%
%
% \end{itemize}
%
% \end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{T2: Disagreement at Calibration Points}
%
%
%
%
% \begin{figure}[!h]
% \begin{center}
% \hspace*{1.5cm}
% %\vspace*{2.5cm}
% {\includegraphics[width=8.0cm]{/home/jeremic/tex/works/Conferences/2009/GheoMat/VandV_01/T2_01.jpg}}
% \hspace*{1.5cm}
% %\vspace*{5.0cm}
% \end{center}
% \end{figure}
%
%
% % \begin{itemize}
% %
% %
% %
% % \end{itemize}
%
% \end{frame}
%
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{T2: Stage 14, Interpretation of Data }
% %
% %
% %
% %
% % \begin{figure}[!h]
% % \begin{center}
% % %\vspace*{2.5cm}
% % \vspace*{1.0cm}
% % \hspace*{1.5cm}
% % {\includegraphics[width=8.0cm]{/home/jeremic/tex/works/Conferences/2009/GheoMat/VandV_01/T2_02.jpg}}
% % \hspace*{1.5cm}
% % \vspace*{1.5cm}
% % %\vspace*{5.0cm}
% % \end{center}
% % \end{figure}
% %
% % %
% % % % \begin{itemize}
% % % %
% % % %
% % %
% % % \end{itemize}
% %
% % \end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
%
%
\section{ESSI Simulator System}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{System Components}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Desirable Modeling and Simulation Capabilities}
\begin{itemize}
\item Body (SH, SV, P) and Surface (Rayleigh, Love, etc) seismic motions modeling
and their input into finite element models
\item Elasticplastic modeling of dry and saturated soil/rock behavior beneath
foundations
\item Elasticplastic modeling of soil/rock (limited data)
\item Soil/rock  foundation contact zone modeling (for dry and saturated
conditions)
\item Verification and Validation suite
\item High performance, parallel simulation using dynamic domain
decomposition (Plastic Domain Decomposition) for elasticplastic simulations
\item Probabilistic elastoplasticity and stochastic elasticplastic finite
element methods
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{ESSI Simulator System}
\begin{itemize}
\item {\bf The ESSIProgram} is a 3D, nonlinear, time domain,
parallel finite element program specifically developed for
HiFi modeling and simulation of Earthquake Soil/Rock Structure
Interaction problems for NPPs on ESSIComputer. \
%The NRC ESSI Program is based on
%a number of public domain numerical libraries developed at UCD as well as those
%available on the web, that are compiled and linked together to form the
%executable program (NRCESSIProgram). Significant effort is devoted to development
%of verification and validation procedures, as well as on development of
%extensive documentation. NRCESSIProgram is in public domain and is licensed
%through the Lesser GPL.
%\vspace*{0.3cm}
\vspace*{0.1cm}
\item {\bf The ESSIComputer} is a distributed memory
parallel computer, a cluster of clusters with multiple performance
processors and multiple performance networks.
%Compute nodes are Shared Memory Parallel
%(SMP) computers, that are connected, using high speed network(s), into a
%Distributed Memory Parallel (DMP) computer.
%\vspace*{0.3cm}
\vspace*{0.1cm}
\item {\bf The ESSINotes} represent a hypertext
documentation system
%(Theory and Formulation, Software and Hardware, Verification and Validation, and
%Case Studies and Practical Examples)
detailing modeling and simulation of NPP ESSI
problems.
%
%the
%NRCESSIProgram code API (application Programming Interface) and DSLs (Domain
%Specific Language).
%%NRCESSINotes, developed by Boris Jeremic and collaborators, are in public
%domain
%%and are licensed under a Creative Commons AttributionShareAlike 3.0 Unported
%%License.
%
%\vspace*{0.3cm}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{ESSI Simulator Program}
\begin{itemize}
%\vspace*{0.2cm}
\item Based on a Collection of Useful Libraries (modular, portable)
\vspace*{0.1cm}
\item Library centric software design
\vspace*{0.1cm}
\item Solids (dry, saturated), beams, shells, contacts, elastic or elasticplastic
\vspace*{0.1cm}
\item Various public domain licenses (GPL, LGPL, BSD, CC)
%\vspace*{0.3cm}
\vspace*{0.1cm}
\item Verification (extensive) and Validation (not much)
\vspace*{0.1cm}
\item Program documentation (part of ESSI Notes)
\vspace*{0.1cm}
\item Target users: USNRC staff, CNSC staff, IAEA, LBNL, INL, DOE,
professional practice collaborators, expert users
%\item Sources will be available through
%{\bf
%\url{http://nrcessisimulator.info}}
\end{itemize}
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Collection of Useful Libraries (Modeling Part)}
%
% \begin{itemize}
%
%
% \vspace*{0.2cm}
% \item Template3DEP libraries for elastic and elasticplastic
% computations (UCD, CC)
%
% \vspace*{0.2cm}
% \item FEMTools finite element libraries provide
% finite elements (solids,
% beams, shells, contacts/isolators, seismic input) (UCD, UCB, CU, CC)
%
%
% \vspace*{0.2cm}
% \item Loading, staged, self weight, service loads, seismic loads
% (the Domain Reduction Method, analytic input
% (incoming/outgoing) of 3D, inclined, uncorrelated seismic motions)
% (UCD, CC)
%
% \vspace*{0.2cm}
% \item Domain Specific Language for input (UCD, CC)
%
%
%
% \end{itemize}
%
% \end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Collection of Useful Libraries (Simulation Part)}
%
% \begin{itemize}
%
%
%
%
%
% \vspace*{0.12cm}
% \item Plastic Domain Decomposition (PDD) for parallel computing (UCD, CC)
%
%
% \vspace*{0.12cm}
% \item PETSc (ANL, GPLlike) and UMFPACK (UF, GPL) solvers
%
%
% \vspace*{0.12cm}
% \item Modified OpenSees Services (MOSS) for managing the finite
% element domain (UCD, CC; UCB, GPL?)
%
% \vspace*{0.12cm}
% \item nDarray (UCD, CC), LTensor (CIMEC, GPL),
% BLAS (UTK, GPL) for lower level
% computational tasks,
%
% \vspace*{0.12cm}
% \item Message Passing Interface (MPI, openMPI, new BSD license)
%
% \end{itemize}
%
% \end{frame}
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile]
\frametitle{ESSI Simulator Computer}
A distributed memory parallel (DMP) computer
designed for high performance,
parallel finite element simulations
\begin{itemize}
%\vspace*{0.1cm}
\item Multiple performance CPUs \\
and Networks
%\vspace*{0.1cm}
\item Most costperformance \\
effective
%\vspace*{0.1cm}
\item Source compatibility with \\
any DMP supercomputer
%\vspace*{0.1cm}
\item Current systems: 208CPUs, \\
and 40CPUs (8+32) and \\
160CPUs (8x5+2x16+24+64)...
%%\vspace*{0.1cm}
% \item Near future: 784 CPUs
\end{itemize}
\vspace*{4.5cm}
\begin{flushright}
%\hspace*{0.5cm}
\includegraphics[width=5.0cm]{/home/jeremic/public_html/NRC_ESSI_Simulator/NRC_ESSI_Computer/photos/IMG_2607.JPG}
%\includegraphics[width=6.0cm]{/home/jeremic/public_html/NRC_ESSI_Simulator/NRC_ESSI_Computer/photos/IMG_2609.JPG}
%\includegraphics[width=8.0cm]{/home/jeremic/public_html/NRC_ESSI_Simulator/NRC_ESSI_Computer/photos/IMG_2611.JPG}
\end{flushright}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %\begin{frame}[fragile]
% % \frametitle{NRC ESSI Simulator Version December 2010}
% %
% %
% %\begin{itemize}
% %\item Operating System: Linux Fedora Core 14.
% %
% %\item Kernel: \verb2.6.35.1074.fc14.x86_64
% %
% %\item Compute Nodes (two):
% %
% % \begin{itemize}
% % \item CPU: 2 $\times$ Intel Xeon E5620
% % Westmere 2.4 GHz Quad Core (8 threads)
% %
% % \item RAM: 6 $\times$ 4GB DDR3 1333 MHz ECC/Registered Memory (24GB
% % Total Memory)
% %
% % \item Disk: 8 $\times$ 500 GB Seagate Constellation ES 3.5" SATA/300
% % (Linux Software RAID10)
% %
% % \end{itemize}
% %
% %\item Network: single GigaBit
% %\end{itemize}
% %
% %
% %\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}[fragile]
% \frametitle{ESSI Computer Version April 2012}
%
%
% Operating System: Ubuntu
%
% Kernel: 3.2
%
% {\bf Controller:} 1 node + {\bf Compute:} 8 Nodes
% \begin{itemize}
% \item CPU: 2 x 12 cores Opteron 6234 = 24 cores
%
% \item RAM: 32GB (8 x 4GB)
%
%
% \item NICs:
% \begin{itemize}
% \item GigaBit: Intel 82576 (Controller)
% \item InfiniBand: ConnectX2 QDR IB 40Gb/s (Controller+Compute)
% \end{itemize}
%
% \item Disk: 8 $\times$ 2TB Toshiba MK2002TSKB (Controller)
% \item Disk: 1TB Toshiba MK1002TSKB (Compute)
%
% \end{itemize}
%
% Network (dual):
% \begin{itemize}
%
% \item GigaBit: HP ProCurve Switch 181048G 48 Port
%
%
% \item InfiniBand:: Mellanox MIS5030Q1SFCA 36port QDR
%
%
% \end{itemize}
%
%
% \end{frame}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{ESSI Simulator Notes}
\begin{itemize}
\item A hypertext documentation system describing in detail modeling and
simulations of NPP ESSI problems
\begin{itemize}
\item Theoretical and Computational Formulations (FEM, ELPL, Static
and Dynamic solution, Parallel Computing)
\item Software and Hardware Platform Design (OO Design, Library centric
design, API, DSL, Software Build Process, Hardware Platform)
\item Verification and Validation (code V, Components V, Static and
Dynamic V, Wave Propagation V)
\item Application to Practical Nuclear Power Plant Earthquake
Soil/Rock Structure Interaction Problems (ESSI with 3D, inclined,
uncorrelated seismic waves, ESSI with foundation slip, Isolators)
\end{itemize}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Modeling and Simulation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
% \frametitle{Seismic Energy Dissipation for \underline{Soil}FoundationStructure Systems}
\frametitle{ESSI: High Fidelity Modeling}
% \frametitle{Seismic Energy Dissipation for
% \underline{Soil}FoundationStructure Systems}
\begin{itemize}
\item Seismic energy influx, {body and surface waves, 3D, inclined}
% $E_{flux} = \rho A c \int_0^t \dot{u}_i^2 dt$ (Aki \& Richards)
\vspace*{0.1cm}
\item Mechanical dissipation outside of SSI domain:
\begin{itemize}
\item {Radiation} of reflected waves
\item {Radiation} of oscillating SSI system
\end{itemize}
\vspace*{0.1cm}
\item Mechanical dissipation inside SSI domain:
\begin{itemize}
\item {Plasticity} of soil/rock subdomain
\item {Viscous coupling} of porous solid with pore fluid (air,
water)
\item {Plasticity} and viscosity of foundation  soil/rock contact
\item Plasticity/damage of the structure
\item Viscous coupling of structure/foundation with fluids
% \item potential and kinetic energy
% \item[] potential $\leftarrow \! \! \! \! \! \! \rightarrow$ kinetic energy
\end{itemize}
\vspace*{0.1cm}
% \item Numerical energy dissipation (numerical damping/production and period errors)
% \item Numerical energy dissipation (damping/production)
\item Numerical energy dissipation/production
\end{itemize}
%
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{ESSI: High Performance, Parallel Computing}
\begin{itemize}
\item The ESSI Program can be used in both sequential and
parallel modes
\vspace*{0.2cm}
\item For high fidelity models, parallel is really the only option
\vspace*{0.2cm}
\item High performance, parallel computing using
Plastic Domain Decomposition
Method, for elasticplastic computations (dynamic computational load balancing)
\vspace*{0.2cm}
\item Developed for multiple/variable capability CPUs and
networks (DMP and
SMPs)
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{ESSI: Finite Elements}
% \begin{itemize}
%
% \item Linear and nonlinear truss element
%
% \item Linear and nonlinear beam (disp. based), variable BC el.
%
% \item Linear shell Triangle and Quad with drilling DOFs
%
% %\item Linear and nonlinear thick shell (bricks)
%
% \item Single phase solid bricks (8, 20, 27, 820, 827 nodes)
%
% \item Two phase (fully coupled, porous solid, pore fluid) solid bricks (8 and
% 20 node: $upU$, $up$)
%
% \item Dry friction slip and gap element
%
% \item Saturated gap and slip element
%
% \item Seismic isolator (latex rubber, neoprene rubber, rubber with lead core,
% friction pendulum)
%
% \end{itemize}
% \end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{ESSI: Material Models for Solids and Structures}
%
% \begin{itemize}
% \item Small deformation elastic: linear, nonlinear isotropic, cross anisotropic
%
% \vspace*{0.3cm}
% \item Small deformation elasticPlastic: von Mises,
% DruckerPrager, CamClay, Rounded MohrCoulomb, Parabolic Leon,
% SaniSand2004, SaniSand2008, SaniClay, Pisan{\`o}; Gens normal contact and
% Coulomb shear contact model; 1D concrete and steel models
%
% %\vspace*{0.3cm}
% % \item Isotropic and kinematic (translational and rotational) hardening
%
%
% \vspace*{0.3cm}
% \item Large deformation elastic and elasticplastic: Ogden, neoHookean,
% MooneyRivlin, Logarithmic, SimoPister, von Mises, DruckerPrager
%
% \end{itemize}
%
% %\vspace*{2.0cm}
% \end{frame}
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{ESSI: Earthquake Ground Motions}
%
%
%
% Realistic earthquake ground motions
% \begin{itemize}
% %\vspace*{0.1cm}
% \item Body: P and S waves
% %\vspace*{0.1cm}
% \item Surface: Rayleigh, Love waves, etc.
% %\vspace*{0.1cm}
% \item Lack of correlation (incoherence)
% %\vspace*{0.1cm}
% \item Inclined waves
% %\vspace*{0.1cm}
% \item 3D waves
%
% \item Seismic input: Domain Reduction Method (Bielak et al.)
%
% \begin{itemize}
% \item Dynamically consistent replacement for a seismic source
% \item Only outgoing waves are from dynamics of the structure
% \item Material can be elasticplastic
% \item All seismic waves (body, surface...) are properly modeled
% \end{itemize}
%
% %\vspace*{0.1cm}
% % \item Earthquake energy dissipation
%
% \end{itemize}
%
%
%
%
%
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Body (P, S) and Surface (Rayleigh, Love) Waves}
%
%
% \vspace*{0.3cm}
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=2.5cm, angle=45]{/home/jeremic/tex/works/consulting/2010/CanadianNuclearSafetyComission/Presentation/P_body_wave.jpeg}
% \includegraphics[width=2.5cm, angle=45]{/home/jeremic/tex/works/consulting/2010/CanadianNuclearSafetyComission/Presentation/S_body_wave.jpeg}
% \vspace*{0.7cm}
% \\
% \includegraphics[width=3cm]{/home/jeremic/tex/works/consulting/2010/CanadianNuclearSafetyComission/Presentation/Rayleigh_surface_wave.jpeg}
% \includegraphics[width=3cm]{/home/jeremic/tex/works/consulting/2010/CanadianNuclearSafetyComission/Presentation/Love_surface_wave.jpeg}
% %\caption{\label{Love_surface_wave} Visualization of propagation of a Love
% %surface seismic wave (illustrations are from MTU web site).}
% \end{center}
% \end{figure}
%
% \end{frame}
%
%
%
%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame}
%  \frametitle{Spatial Variability (Incoherence, Lack of Correlation)}
% 
%  Incoherence $\rightarrow$ frequency domain
% 
%  \vspace*{0.2cm}
% 
%  Lack of Correlation $\rightarrow$ time domain
% 
% 
%  \vspace*{0.5cm}
% 
%  \begin{itemize}
%  \item Attenuation effects
%  \item Wave passage effects
%  \item Extended source effects
%  \item Scattering effects
%  \item Variable seismic energy dissipation
%  \end{itemize}
% 
%  %\begin{figure}[!htb]
%  %\begin{center}
%  \vspace*{3.5cm}
%  \hspace*{5.5cm}
%  \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_25May2011/Lack_of_Correlation_5_points.pdf}
%  %\caption{\label{LC} Four main sources contributing to the lack of correlation of
%  %seismic waves as measured at two observation points.}
%  %\end{center}
%  %\end{figure}
%  %
%  %A number of models available (Abrahamson...)
%  %
%  \end{frame}
% 
% 
%  % out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  % out \begin{frame}
% out \frametitle{Attenuation Effects}
% out
% out
% out Responsible for change in amplitude and phase of seismic motions
% out due to the distance between observation points and losses (damping, energy dissipation) that
% out seismic wave experiences along that distance. This is a significant source of lack of correlation
% out for long structures (bridges), however for NPPs it is not of much significance.
% out
% out
% out %\begin{figure}[!htb]
% out \begin{center}
% out %\vspace*{2cm}
% out %\hspace*{5.5cm}
% out \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_25May2011/Lack_of_Correlation_5_points.pdf}
% out %\caption{\label{LC} Four main sources contributing to the lack of correlation of
% out %seismic waves as measured at two observation points.}
% out \end{center}
% out %\end{figure}
% out
% out
% out
% out \end{frame}
% out
% out
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out \begin{frame}
% out \frametitle{Wave Passage Effects}
% out
% out Contribute to lack of correlation due to difference in
% out recorded wave field at two location points as the (surface) wave travels,
% out propagates from the first to second point.
% out
% out \begin{center}
% out %\hspace*{5.5cm}
% out \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_25May2011/Lack_of_Correlation_5_points.pdf}
% out \end{center}
% out
% out
% out \end{frame}
% out
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out \begin{frame}
% out \frametitle{Extended Source Effects}
% out
% out Contribute to lack of correlation by creating a complex wave source
% out field, as the fault ruptures, rupture propagates and generate seismic sources along the fault.
% out Seismic energy is thus emitted from different points (along the rupturing fault) and will have
% out different travel path and timing as it makes it observation points.
% out
% out
% out \begin{center}
% out %\hspace*{5.5cm}
% out \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_25May2011/Lack_of_Correlation_5_points.pdf}
% out \end{center}
% out
% out \end{frame}
% out
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out \begin{frame}
% out \frametitle{Scattering Effects}
% out
% out Responsible to lack of correlation by creating a
% out scattered wave field.
% out Scattering is due to (unknown or not known enough) subsurface geologic features
% out that contribute to (elastic) modification of the wave field.
% out
% out
% out \begin{center}
% out %\hspace*{5.5cm}
% out \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_25May2011/Lack_of_Correlation_5_points.pdf}
% out \end{center}
% out
% out
% out
% out \end{frame}
% out
% out
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out \begin{frame}
% out \frametitle{Variable Seismic Energy Dissipation}
% out
% out Contribute to variability of seismic motions by bending seismic waves as they
% out pass through inelastic soil/rock.
% out Variable seismic energy dissipation is due to (unknown or not known enough)
% out subsurface geologic features that contribute to (inelastic, elasticplastic)
% out modification of the wave field.
% out
% out
% out \begin{center}
% out %\hspace*{5.5cm}
% out \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_25May2011/Lack_of_Correlation_5_points.pdf}
% out \end{center}
% out
% out
% out
% out
% out \end{frame}
% out
% out
% out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% out \begin{frame}
% out \frametitle{Modeling Lack of Correlation (Incoherence)}
% out
% out \begin{itemize}
% out
% out \item A number of models available (Abrahamson...)
% out
% out \item Most of the models (all) are based on a (very) limited data set from Lotung, Pinyon Flat...
% out
% out \item Most of the models (all) are based on hard rock data ($V_s > 2600$m/s)
% out
% out \item Most of the models (all) can produce statistically significant number of
% out motions, yet only few are used (destroying the model statistical assumptions)
% out
% out
% out \item Ergodic assumption must be made in order to extrapolate those models
% out (data) to other parts of the USA (world)
% out
% out %\item Extrapolations can be (are) dangerous
% out
% out
% out \end{itemize}
% out
% out
% out \end{frame}
% out
% out
% out
% out
% out
% out
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame}
%  \frametitle{Seismic Input}
% 
% 
%  %\begin{itemize}
% 
%  % \item
%  The Domain Reduction Method \\
%  (Bielak et al.): \\
%  The effective force $P^{eff}$ \\
%  is a dynamically consistent \\
%  replacement for the dynamic \\
%  source forces $P_{e}$
% 
%  % \end{itemize}
% 
%  \begin{eqnarray}
%  P^{eff} = \left\{\begin{array}{c} P^{eff}_i \\ P^{eff}_b \\ P^{eff}_e \end{array}\right\}
%  = \left\{\begin{array}{c} 0 \\ M^{\Omega+}_{be} \ddot{u}^0_eK^{\Omega+}_{be}u^0_e
%  \\ M^{\Omega+}_{eb}\ddot{u}^0_b+K^{\Omega+}_{eb}u^0_b\end{array}\right\}
%  \nonumber
%  \label{DRMeq09}
%  \end{eqnarray}
%  %
% 
%  \begin{figure}[!h]
%  \begin{flushright}
%  %\vspace*{0.50cm}
%  %\begin{center}
%  %\hspace*{1cm}
%  \vspace*{6.90cm}
%  {\includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2010/NRCLBLProjectReviewMeeting_21_22_Sept_2010/DRM05NPP.pdf}}
%  %\vspace*{5.50cm}
%  %\hspace*{1cm}
%  %\vspace*{2.50cm}
%  %\end{center}
%  %\vspace*{0.3cm}
%  \end{flushright}
%  \end{figure}
% 
% 
% 
% 
%  \end{frame}
% 
% 
% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{DRM}
%
%
%
%
%
% \begin{itemize}
% %\vspace*{0.2cm}
% \item Seismic forces $P_e$ replaced by $P^{eff}$
%
% %\vspace*{0.2cm}
% \item $P^{eff}$ applied only to a single \\
% layer of elements next to $\Gamma$.
% %\vspace*{0.2cm}
% \item The only outgoing waves are \\
% from dynamics of the NPP
% %\vspace*{0.2cm}
% \item Material inside $\Omega$ \\
% can be elasticplastic
%
% \item All types of seismic waves\\
% (body, surface...) are \\
% properly modeled
%
%
% % \item The only input wave field is the one for the nodes of this layer of elements.
% \end{itemize}
%
% \begin{figure}[!h]
% \begin{flushright}
% %\vspace*{0.50cm}
% %\begin{center}
% %\hspace*{1cm}
% \vspace*{4.50cm}
% {\includegraphics[width=5.8cm]{/home/jeremic/tex/works/Conferences/2010/NRCLBLProjectReviewMeeting_21_22_Sept_2010/DRM05NPP.pdf}}
% %\vspace*{5.50cm}
% \hspace*{0.8cm}
% %\vspace*{2.50cm}
% %\end{center}
% %\vspace*{0.3cm}
% \end{flushright}
% \end{figure}
%
%
% \end{frame}
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Verification and Validation Suite}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Verification, Validation (V\&V) and Prediction}
\begin{itemize}
\item Verification: the process of determining that a model
implementation accurately represents the developer's conceptual description
and specification. Mathematics issue. {\it Verification provides evidence that the
model is solved correctly.}
\item Validation: The process of determining the degree to which a
model is accurate representation of the real world from the perspective of
the intended uses of the model. Physics issue. {\it Validation provides
evidence that the correct model is solved.}
\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
\end{itemize}
%
%\item Models available (some now, some later)
%\vspace*{2.0cm}
\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=6.5cm]{/home/jeremic/tex/works/Conferences/2011/USNCCM11_Minneapolis/Coupled/Present/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}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Importance of V\&V}
%
%
% \begin{itemize}
%
% % \vspace*{2.0truecm}
% \item V \& V procedures are the primary means of assessing accuracy in
% modeling and computational simulations
%
% \vspace*{0.5truecm}
% \item V \& V procedures are the tools with which we build confidence and
% credibility in modeling and computational simulations
%
%
% \end{itemize}
%
%
% \end{frame}
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{V\&V for ESSI Modeling and Simulations}
\begin{itemize}
\vspace*{0.2cm}
\item Code Verification
\vspace*{0.2cm}
\item Material modeling and simulation (elastic, elasticplastic...)
\vspace*{0.2cm}
\item Finite elements (solids, structural, special...)
\vspace*{0.2cm}
\item Solution advancement algorithms (static, dynamic...)
\vspace*{0.2cm}
\item Seismic input and radiation
\vspace*{0.2cm}
\item Finite element model verification
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Constitutive Integration Error Maps}
Normalized error: $ \delta^r = {\sqrt{(\sigma_{ij} 
\sigma^{\ast}_{ij})(\sigma_{ij}  \sigma^{\ast}_{ij})}}/
{\sqrt{\sigma^{0}_{pq}\sigma^{0}_{pq}}}$
SaniSand2004, rot. kinematic hardening, bounding surface:
%%%
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width = 10cm]{/home/jeremic/tex/works/Thesis/ZhaoCheng/thesis/file/NewTemplate3Dep/ErrorMap/DM_ConstantE/errMap_normalized_DM.pdf}
%\includegraphics[width = 10cm]{./Present06_figs/errMap_normalized_DM.pdf}
\end{center}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Material Model Validation (SanSand2004)}
\begin{figure}[!htbp]
\hspace*{0.4cm}
%\begin{center}
\includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/Coupled/Present/DMvalidation01a.jpg}
%\includegraphics[width=5cm]{./Present06_figs/DMvalidation01a.jpg}
\hspace*{0.5cm}
\includegraphics[width=5.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/Coupled/Present/DMvalidation02a.jpg}
%\includegraphics[width=5.2cm]{./Present06_figs/DMvalidation02a.jpg}
\hspace*{0.4cm}
%\end{center}
\end{figure}
%
%
%
%\item Models available (some now, some later)
%\vspace*{2.0cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Verification ANDES Shell: Static}
\begin{center}
\includegraphics[width=0.55\textwidth]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Jose_fix/presentation/Present06_figs/fig1_static_tests.pdf}
% \includegraphics[width=0.6\textwidth]{Present06_figs/fig1_static_tests.pdf}
\\
\includegraphics[width=0.55\textwidth]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Jose_fix/presentation/Present06_figs/fig2_static_tests.pdf}
% \includegraphics[width=0.6\textwidth]{Present06_figs/fig2_static_tests.pdf}
\end{center}
Material parameters chosen such that the exact solution is $u_z = 100.000$ $T=1.0\,\mathrm{s}$.
Nz = 2, $u_z = 96.212$ ; Nz = 7, $u_z = 100.096$
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Verification ANDES Shell: Dynamic}
\begin{center}
\includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Jose_fix/presentation/Present06_figs/Test_shell_andes_1_free_vibration_EigenMode1.png}
% \includegraphics[width=0.7\textwidth]{Present06_figs/Test_shell_andes_1_free_vibration_EigenMode1.png}
\\
\includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Jose_fix/presentation/Present06_figs/Test_shell_andes_2_free_vibration_EigenMode1.png}
% \includegraphics[width=0.7\textwidth]{Present06_figs/Test_shell_andes_2_free_vibration_EigenMode1.png}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Verification ANDES Shell: Dynamic}
%
% Comparison to MITC4 shell and simple beam element for inplane dynamic bending.
%
% \begin{center}
% % \includegraphics[width=\textwidth]{./_Jose_Files/beam_vs_mitc4_vs_andes.pdf}
% \includegraphics[width=\textwidth]{../../_Jose_Files/beam_vs_mitc4_vs_andes.pdf}
% % beam_vs_mitc4_vs_andes.pdf: 576x432 pixel, 72dpi, 20.32x15.24 cm, bb=0 0 576 432
% \end{center}
%
%
% \end{frame}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Verification for 27 Node Brick}
%Lenght=10m, Force=9N, E=100000Pa, $I=\frac{1}{12}m^4$
\begin{eqnarray}
d = \frac{PL^3}{3EI} = \frac{9\rm{N} \times 1000\rm{m}^3}{3 \times 100000\rm{Pa}
\times \frac{1}{12}\rm{m}^4}=0.36m
\nonumber
\end{eqnarray}
\begin{flushright}
\includegraphics[width=0.3\textwidth]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Verification_and_Validation_for_Static_and_Dynamic_Behavior_of_Single_Phase_Solid_Elements/cantilever_27nodebrick_10elements.png}
\includegraphics[width=0.3\textwidth]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Verification_and_Validation_for_Static_and_Dynamic_Behavior_of_Single_Phase_Solid_Elements/cantilever_27nodebrick_2elements.png}
\includegraphics[width=0.3\textwidth]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Verification_and_Validation_for_Static_and_Dynamic_Behavior_of_Single_Phase_Solid_Elements/cantilever_27nodebrick_1element.png}
%\includegraphics[width=0.6\textwidth]{Present06_figs/cantilever_27nodebrick_10elements.png}
\end{flushright}
\vspace*{1cm}
\begin{eqnarray}
errors: \hspace{1cm} 0.47\% \hspace{2cm} 3.96\% \hspace{2cm} 22\%
\nonumber
\end{eqnarray}
for nodal offset: 40\% error: 2\%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Shock Wave Propagation, Step Displacement}
\begin{figure}[!hbpt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=0.8\textwidth]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO/GAJO_Figures/numerical_model.pdf}
\end{center}
%\caption{\label{Fig1001}{(a) The representative semifinite soil column
%subjected to a step vertical displacement equal to $1.0\times10^{3}cm$ at the
%surface, (b) the finite element mesh and the applied boundary conditions used
%for the numerical modeling and (c) the time history of the vertical displacement
%applied at the top nodes of the mesh, both to the solid and fluid phases
%(Heaviside function). }}
\end{sffamily}
\end{Large}
\end{figure}
%
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Shock Wave Propagation: Step Displacement}
%
%
%
%
%
% \vspace*{0.5cm}
% \begin{figure}[!hbpt]
% \begin{center}
% \hspace*{0.8cm}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/solid_k108_large.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/solid_k108_close.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/fluid_k108_large.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/fluid_k108_close.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/solid_k106_large.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/fluid_k106_large.pdf}
% \\
% \vspace*{0.1cm}
% \hspace*{0.8cm}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/fluid_k106_close.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/solid_k105_large.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/solid_k105_close.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/fluid_k105_large.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/fluid_k105_close.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/comparison_large_solid.pdf}
% \\
% \vspace*{0.1cm}
% \hspace*{0.8cm}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/comparison_close_solid.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/comparison_large.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/comparison_close.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/solid_k_108_large.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/solid_k_108_close.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/fluid_k_108_large.pdf}
% \\
% \vspace*{0.1cm}
% \hspace*{0.8cm}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/fluid_k_108_close.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/solid_k_106_large.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/solid_k_108_close.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/fluid_k_106_large.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/fluid_k_108_close.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/solid_k_105_large.pdf}
% \\
% \vspace*{0.1cm}
% \hspace*{0.8cm}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/solid_k_105_close.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/fluid_k_105_large.pdf}
% \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/fluid_k_105_close.pdf}
% %\caption{\label{Fig1}{Time histories of solid displacement due to longitudinal wave at the depth of 1 cm below the surface. Comparison between numerical results (FEM) and analytical solution by \cite{Gajo1995b} for the case of viscous coupling, $k$, equal to $10^{8} cm^3s/g$. Two different sets of Newmark parameters were used for the numerical analysis.}}
% \end{center}
% \end{figure}
%
% \end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Shock Wave Propagation: Porous Solid, Pore Fluid}
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=5cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/comparison_large_solid.pdf}
\hfill
\includegraphics[width=5cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/comparison_large.pdf}
%\caption{\label{Fig1}{Time histories of solid displacement due to longitudinal wave at the depth of 1 cm below the surface. Comparison between numerical results (FEM) and analytical solution by \cite{Gajo1995b} for the case of viscous coupling, $k$, equal to $10^{8} cm^3s/g$. Two different sets of Newmark parameters were used for the numerical analysis.}}
\end{center}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Verification for Dry Contact Element: \\
Truss Model, Normal Displacement}
\vspace*{0.5cm}
\begin{figure}[htbp]
\centering
\includegraphics[width=4cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Verification_and_Validation_for_Static_and_Dynamic_Behavior_of_Special_Elements/Contact_Element_MM_truss_u_normal.pdf}
%\includegraphics[width=4cm]{Present06_figs/Contact_Element_MM_truss_u_normal.pdf}
\hspace*{5mm}
\includegraphics[width=8cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Verification_and_Validation_for_Static_and_Dynamic_Behavior_of_Special_Elements/Contact_Element_MM_truss_u_normal_results.pdf}
%\includegraphics[width=8cm]{Present06_figs/Contact_Element_MM_truss_u_normal_results.pdf}
\end{figure}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Solution Advancement \\
(Newmark, HilberHughesTaylor)}
\vspace*{1.0cm}
\begin{itemize}
% \item 2DOF system periods $T_1 = 4.0\,\mathrm{s}$ and $T_2 = 1.0\,\mathrm{s}$.
\item Variable integration \\
steps sizes, \\
parameters ($\alpha$, $\beta$, $\gamma$)
\item Compare with \\
theoretical \\
algorithmic damping \\
(spectral radius) \\
and period shift
\end{itemize}
\vspace*{5.5cm}
\begin{flushright}
% \includegraphics[width=\textwidth]{./_Jose_Files/integrators/p2a_newmark_08errors.pdf}
\hspace*{4.5cm}
\includegraphics[width=7.5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Jose_fix/presentation/Present06_figs/p2a_newmark_08errors.pdf}
% p2a_newmark_05top.pdf: 1296x521 pixel, 72dpi, 45.72x18.38 cm, bb=0 0 1296 521
\end{flushright}
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\
% \begin{frame}
% \frametitle{Newmark Verification}
% \begin{center}
% % \includegraphics[width=\textwidth]{./_Jose_Files/integrators/p2a_newmark_08top.pdf}
% \includegraphics[width=12cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Jose_fix/presentation/Present06_figs/p2a_newmark_08top.pdf}
% % p2a_newmark_05top.pdf: 1296x521 pixel, 72dpi, 45.72x18.38 cm, bb=0 0 1296 521
% \end{center}
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\
% \begin{frame}
% \frametitle{Newmark Verification}
%
% \vspace*{0.1cm}
% \begin{center}
% % \includegraphics[width=\textwidth]{./_Jose_Files/integrators/p2a_newmark_08errors.pdf}
% \includegraphics[width=8.5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Jose_fix/presentation/Present06_figs/p2a_newmark_08errors.pdf}
% % p2a_newmark_05top.pdf: 1296x521 pixel, 72dpi, 45.72x18.38 cm, bb=0 0 1296 521
% \end{center}
% \end{frame}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Seismic Body and Surface Waves}
\begin{itemize}
\item Both body (P, SV and SH) and surface (Rayleigh, Love, etc.) waves
are present
\vspace*{1mm}
\item Surface waves carry most seismic energy
\vspace*{1mm}
\item Analytic (Aki and Richards, Trifunac and Lee, Hisada et
al., fk, etc.) and numerically generated, 3D, inclined (plane) body and
surface waves are used in tests
\vspace*{1mm}
\item Seismic moment from a point source at $2$km depth used
\vspace*{1mm}
\item Stress drop at the source: Ricker and/or Ormsby wavelets
\vspace*{1mm}
\item Seismic input into FE model using the DRM (Bielak at al.)
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Plane Wave Model}
\vspace*{1cm}
\begin{figure}[!h]
\begin{flushright}
\includegraphics[width=4cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Verification_and_Validation_for_Seismic_Wave_Propagation_Problems/tex_works_Thesis_NimaTafazzoli_wave_propagation_figs_DRMModel.pdf}
\label{fig:DRMModel}
\end{flushright}
\end{figure}
\vspace*{1.5cm}
\begin{figure}[H]
\begin{center}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2011/NRC_LBNL_Review_Panel_Sept2011/2D_faul_slip_model.pdf}
\end{center}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Seismic Source Mechanics}
\vspace*{0.5cm}
Stress drop, Ormsby wavelet
\vspace*{1cm}
\begin{figure}[H]
\begin{flushright}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Conferences/2011/NRC_LBNL_Review_Panel_Sept2011/Seismic_source_moment_couple.pdf}
\end{flushright}
\end{figure}
\vspace*{1.9cm}
\hspace*{1cm}
\begin{figure}[H]
\begin{center}
\hspace*{0.4cm}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/FaultModel_7seconds/xz_TimeHistory/3000_3000_x_displacement.pdf}
\hspace*{0.4cm}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/FaultModel_7seconds/xz_FFT/3000_3000_x_displacement_FFT.pdf}
\end{center}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Middle (Structure Location) Plane, Top 2km}
\vspace*{0.4cm}
\begin{figure}[H]
\begin{flushright}
\includegraphics[width=3cm]{/home/jeremic/tex/works/Conferences/2011/NRC_LBNL_Review_Panel_Sept2011/2D_faul_slip_model_MIDDLE.pdf}
\end{flushright}
\end{figure}
\vspace*{2.0cm}
\begin{figure}[H]
\begin{center}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/FreeFieldInclinedMotionModels/Ormsby/middle_top2000/middle_acceleration_x.pdf}
\hspace*{0.5cm}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/FreeFieldInclinedMotionModels/Ormsby/middle_top2000/middle_acceleration_z.pdf}
\end{center}
\end{figure}
\vspace*{0.90cm}
{horizontal accelerations \hfill vertical accelerations}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Verification: Displacements, Top Middle Point }
% \begin{itemize}
% \item
% \end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!htbp]
\begin{center}
\begin{tabular}{ccc}
%\hline
(X)
&
(Z)
\\
\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/ricker_2km/top_middle_comparison_disp_x.pdf}
&
\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/ricker_2km/top_middle_comparison_disp_z.pdf}
&
\end{tabular}
%\caption{Comparison of displacements for top middle point using Ricker wave $(f=1Hz)$ as an input motion}
%\label{fig:ricker_acc}
\end{center}
\end{figure}
\end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Verification: Accelerations, Top Middle Point }
% % \begin{itemize}
% % \item
% % \end{itemize}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{figure}[!htbp]
% \begin{center}
% \begin{tabular}{ccc}
% %\hline
% (X)
% &
% (Z)
% \\
% \includegraphics[width=5.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/ricker_2km/top_middle_comparison_accel_x.pdf}
% &
% \includegraphics[width=5.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/ricker_2km/top_middle_comparison_accel_z.pdf}
% &
% \end{tabular}
% %\caption{Comparison of accelerations for top middle point using Ricker wave $(f=1Hz)$ as an input motion}
% %\label{fig:ricker_acc}
% \end{center}
% \end{figure}
%
%
%
%
% \end{frame}
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Verification: Disp. and Acc., Out of DRM }
% \begin{itemize}
% \item
% \end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!htbp]
\begin{center}
\begin{tabular}{ccc}
%\hline
Displacement
&
Acceleration
\\
\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/ricker_2km/10_40_disp_x.pdf}
&
\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/ricker_2km/10_40_accel_x.pdf}
&
\end{tabular}
%\caption{Displacement and acceleration time history for a point outside of DRM layer in (x) direction}
%\label{fig:out_ricker_disp}
\end{center}
\end{figure}
\end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{FE Model}
% \vspace*{3.5cm}
% \begin{figure}
% % \includegraphics[scale=0.35]{Present06_figs/3DModel.pdf}
% \includegraphics[scale=0.35]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Jose_fix/presentation/Present06_figs/3DModel.pdf}
% \end{figure}
% \end{frame}
%
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{FE Model}
% % \begin{figure}
% % % \includegraphics[scale=0.4]{Present06_figs/3DModelxzPlane.pdf}
% % \includegraphics[scale=0.4]{Present06_figs/3DModelxzPlane.pdf}
% % \end{figure}
% % \end{frame}
% %
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Properties}
% \begin{itemize}
% \item Model Properties
% \begin{itemize}
% \item 90 m $\times$ 90 m $\times$ 90 m dimension
% \item Vs1 = 300 m/s, Vs2 = 400 m/s
% \item Poisson's ratio1 = 0.25, Poisson's ratio2 = 0.25
% \item Density1 = 940 kg/m$^3$, Density2 = 990 kg/m$^3$
% \end{itemize}
% \item Input Wave Properties
% \begin{itemize}
% \item Generated using fk program
% \item Variables are chosen to simulate Northridge Earthquake
% \end{itemize}
% \end{itemize}
% \end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{FEM Results, EW Component}
% \vspace*{3.5cm}
% \begin{figure}
% % \includegraphics[scale=0.5]{Present06_figs/90m_Northridge_EW.pdf}
% \includegraphics[scale=0.5]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Jose_fix/presentation/Present06_figs/90m_Northridge_EW.pdf}
% \end{figure}
% \end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{FEM Results, NC Component}
% \vspace*{3.5cm}
% \begin{figure}
% % \includegraphics[scale=0.5]{Present06_figs/90m_Northridge_NS.pdf}
% \includegraphics[scale=0.5]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Jose_fix/presentation/Present06_figs/90m_Northridge_NS.pdf}
% \end{figure}
% \end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{FEM Results, UD Component}
% \vspace*{3.5cm}
% \begin{figure}
% % \includegraphics[scale=0.5]{Present06_figs/90m_Northridge_UD.pdf}
% \includegraphics[scale=0.5]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Jose_fix/presentation/Present06_figs/90m_Northridge_UD.pdf}
% \end{figure}
% \end{frame}
%
%
%
%
%
%
%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Model Verification: Mesh Size Effects}
%
% \begin{itemize}
%
% \item Finite element mesh "filters out" \\
% high frequencies
%
% %\vspace*{0.2cm}
% \item Usual rule of thumb: 1012 elements \\
% needed per wave length
% % (SASSI recommends only 5 ?!)
%
% %
% % \item Maximum grid spacing should not exceed
% % $\Delta h \;\le\; {\lambda}/{10}\;=\;{v}/({10\,f_{max}})$
% % where $v$ is the lowest wave velocity (shear, elasticplastic ?)
% %
% % \item Tests without and with numerical damping, for different element sizes
% %
%
% \item 1D wave propagation model
% %\vspace*{0.2cm}
% \item 3D finite elements (same in 3D)
% %\vspace*{0.2cm}
% \item Motions applied as displacements at the bottom
%
%
% \end{itemize}
%
% %\begin{figure}[H]
% \vspace*{4.0cm}
% \begin{flushright}
% \includegraphics[width=0.7cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_21Nov2011/model01.pdf}
% \end{flushright}
% %\end{figure}
%
% \vspace*{0.4cm}
% \begin{small}
% \begin{table}[!htbp]
% \centering
% % \begin{tabular}{ccccc}
% \begin{tabular}{rm{2.6cm}m{1.5cm}m{1.8cm}m{2.3cm}}
% \hline
% case & model height [m] & $V_s$ [m/s] & El.size [m] & $f_{max}$ (10el) [Hz]\\
% \hline
% %\hline
% 3 & 1000 & 1000 & 10 & 10\\
% %\hline
% 4 & 1000 & 1000 & 20 & 5\\
% %\hline
% 6 & 1000 & 1000 & 50 & 2\\
% \hline
% % \begin{tabular}{m{1.5cm}cm{2.8cm}cm{2.8cm}cm{3.0cm}cm{4.0cm}c}
% % \begin{tabularx}{\linewidth}{ccccc}
% % \begin{tabular*}{0.75\textwidth}{@{\extracolsep{\fill}}ccccc}
% \end{tabular}
% % \end{tabularx}
% \end{table}
% \end{small}
%
%
%
%
% \end{frame}
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Cases 3, 4, and 6, Ormsby Wavelet Input Motions}
%
%
% \begin{figure}[H]
% \begin{center}
% \includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/MeshSize/figs/3_4_6/Input_Displacement.pdf}
% \end{center}
% \end{figure}
%
%
%
% \end{frame}
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Cases 3, 4, and 6, Surface Motions}
%
%
% \begin{figure}[H]
% \begin{center}
% \includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/MeshSize/figs/3_4_6/displacement.pdf}
% \end{center}
% \end{figure}
%
%
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Cases 3, 4, and 6, Input and Surface Motions, FFT}
%
%
% \begin{figure}[H]
% \begin{center}
% \includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/MeshSize/figs/3_4_6/FFT.pdf}
% \end{center}
% \end{figure}
%
%
%
% \end{frame}
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Select Examples: Foundation Slip and Stochastic}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Influence of Inelastic FoundationSoil/Rock Contact on the NPP Response}
\begin{itemize}
\item Soil/rock  foundation interface slip behavior
\vspace*{0.2cm}
\item Changes in Earthquake Soil/Rock Structure Interaction (reduction or
increase in demand)
\vspace*{0.2cm}
\item Dissipation of seismic energy in the slip plane
\vspace*{0.2cm}
\item Passive (and active) base isolation
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Example Model, ESSI with Slip}
%\vspace*{0.5cm}
\begin{figure}[!h]
\begin{center}
{\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2011/NRC_LBNL_Review_Panel_Feb2011/Case_study_model/visit0002.jpeg}}
\end{center}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Morgan Hill Earthquake}
%
% %\vspace*{3.5cm}
% \begin{figure}[H]
% \begin{center}
% \hspace*{0.5cm}
% \includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/MorganHill/acceleration.pdf}
% \hspace*{0.5cm}
% \includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/MorganHill/FFT.pdf}
% \end{center}
% \end{figure}
%
% \end{frame}
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Response to Vertically Propagating Wave, Acc}
%
% \vspace*{0.3cm}
% \begin{figure}[H]
% \begin{center}
% \begin{tabular}{rr}
% %\hline
% %Acceleration on Top (X)
% %&
% %Acceleration on Top (Z)
% %\\
% \mbox{\tiny top X}\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/30_45/top_structure_x_acceleration.pdf}
% &
% \mbox{\tiny top Z}\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/30_45/top_structure_z_acceleration.pdf}
% \\
% %\\
% %% \hline
% %Acceleration at Bottom (X)
% %&
% %Acceleration at Bottom (Z)
% %\\
% \mbox{\tiny bottom X}\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/30_45/bottom_structure_x_acceleration.pdf}
% &
% \mbox{\tiny bottom Z}\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/30_45/bottom_structure_z_acceleration.pdf}
% %\\
% %
% %\hline
% \end{tabular}
% %\caption{Comparison of acceleration time histories of the structure between
% %slipping and noslipping models for Morgan Hill earthquake}
% \label{fig:1d_morgan_acc}
% \end{center}
% \end{figure}
%
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Response to Vertically Propagating Wave, FFT}
%
%
% \vspace*{0.3cm}
% \begin{figure}[H]
% \begin{center}
% \begin{tabular}{rr}
% %%\hline
% %FFT on Top (X)
% %&
% %FFT on Top (Z)
% %\\
% \mbox{\tiny top X}\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/30_45/top_structure_x_acceleration_FFT.pdf}
% &
% \mbox{\tiny top Z}\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/30_45/top_structure_z_acceleration_FFT.pdf}
% %\\
% %\\
% %% \hline
% %FFT at Bottom (X)
% %&
% %FFT at Bottom (Z)
% \\
% \mbox{\tiny bottom X}\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/30_45/bottom_structure_x_acceleration_FFT.pdf}
% &
% \mbox{\tiny bottom Z}\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/30_45/bottom_structure_z_acceleration_FFT.pdf}
% %\\
% %
% %\hline
% \end{tabular}
% %\caption{Comparison of FFT of the acceleration of the structure between
% %slipping and noslipping models for Morgan Hill earthquake}
% \label{fig:1d_morgan_fft}
% \end{center}
% \end{figure}
%
%
% \end{frame}
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Sliding Response}
%
%
% \vspace*{0.1cm}
% \begin{tiny}
% \begin{figure}[H]
% \begin{center}
% \begin{tabular}{ccc}
% %\hline
% $0.5s$
% &
% $0.6s$
% &
% $0.7s$
% \\
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_30_9pieces/slide50.pdf}
% &
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_30_9pieces/slide60.pdf}
% &
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_30_9pieces/slide70.pdf}
% \\
% % \hline
% $0.8s$
% &
% $0.9s$
% &
% $1.0s$
% \\
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_30_9pieces/slide80.pdf}
% &
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_30_9pieces/slide90.pdf}
% &
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_30_9pieces/slide100.pdf}
% \\
% % \hline
% $1.1s$
% &
% $1.2s$
% &
% $1.3s$
% \\
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_30_9pieces/slide110.pdf}
% &
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_30_9pieces/slide120.pdf}
% &
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_30_9pieces/slide130.pdf}
% %\\
% %
% %\hline
% \end{tabular}
% %\caption{Distribution of sliding along the contact interface for Morgan Hill earthquake
% %(gray scale given in meters)}
% \label{fig:1d_morgan_slide_9}
% \end{center}
% \end{figure}
% \end{tiny}
%
%
%
%
%
% \end{frame}
%
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Full 3D (wave at $45^o$) Ricker Wavelet}
\vspace*{0.3cm}
\begin{figure}[H]
\begin{center}
\includegraphics[width=5cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Ricker/acceleration.pdf}
\includegraphics[width=5cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Ricker/FFT.pdf}
\\
\vspace*{0.2cm}
\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/fault_slip_model/FaultSlipModel2km.pdf}
%\caption{Domain to be analyzed for the $1^{st}$ stage of DRM with fault located at an angle of
% $45^{\circ}$ with respect to the top middle point of the model}
\label{fig:FaultSlipModel2km}
\end{center}
\end{figure}
\vspace*{0.5cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Acc. Response for a Full 3D (at $45^\circ$) Ricker Wavelet}
\begin{figure}[H]
\begin{center}
\begin{tabular}{rr}
%\hline
\mbox{\tiny top X}\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/91_97/top_structure_x_acceleration.pdf}
&
\mbox{\tiny top Z}\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/91_97/top_structure_z_acceleration.pdf}
\\
\mbox{\tiny bottom X}\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/91_97/bottom_structure_x_acceleration.pdf}
&
\mbox{\tiny bottom Z}\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/91_97/bottom_structure_z_acceleration.pdf}
\end{tabular}
%\caption{Comparison of acceleration time histories of the structure between
%slipping and noslipping models for Ricker wave}
\label{fig:3d_ricker_acc_1000}
\end{center}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{FFT Response for a Full 3D (at $45^\circ$) Ricker Wavelet}
\begin{figure}[H]
\begin{center}
\begin{tabular}{rr}
\mbox{\tiny top X}\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/91_97/top_structure_x_acceleration_FFT.pdf}
&
\mbox{\tiny top Z}\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/91_97/top_structure_z_acceleration_FFT.pdf}
\\
\mbox{\tiny bottom X}\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/91_97/bottom_structure_x_acceleration_FFT.pdf}
&
\mbox{\tiny bottom Z}\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/91_97/bottom_structure_z_acceleration_FFT.pdf}
\end{tabular}
%\caption{Comparison of FFT of the acceleration of the structure between
%slipping and noslipping models for Ricker wave}
\label{fig:3d_ricker_fft_1000}
\end{center}
\end{figure}
\end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Gaping Response ($45^\circ$ Ricker Wavelet)}
%
% \vspace*{0.1cm}
% \begin{tiny}
% \begin{figure}[H]
% \begin{center}
% \begin{tabular}{ccc}
% %\hline
% $4.5s$
% &
% $4.6s$
% &
% $4.7s$
% \\
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap450.pdf}
% &
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap460.pdf}
% &
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap470.pdf}
% \\
% $4.8s$
% &
% $4.9s$
% &
% $5.0s$
% \\
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap480.pdf}
% &
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap490.pdf}
% &
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap500.pdf}
% \\
% $5.1s$
% &
% $5.2s$
% &
% $5.3s$
% \\
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap510.pdf}
% &
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap520.pdf}
% &
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap530.pdf}
% %\\
% %
% %\hline
% \end{tabular}
% %\caption{Distribution of gap openings along the contact interface for Ricker wave
% %(gray scale given in meters)}
% \label{fig:3d_ricker1000_gap_9}
% \end{center}
% \end{figure}
% \end{tiny}
%
%
%
% \end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Slipping Response ($45^\circ$ Ricker Wavelet)}
%
%
%
%
%
% \vspace*{0.1cm}
% \begin{tiny}
% \begin{figure}[H]
% \begin{center}
% \begin{tabular}{ccc}
% %\hline
% $4.5s$
% &
% $4.6s$
% &
% $4.7s$
% \\
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide450.pdf}
% &
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide460.pdf}
% &
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide470.pdf}
% \\
% $4.8s$
% &
% $4.9s$
% &
% $5.0s$
% \\
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide480.pdf}
% &
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide490.pdf}
% &
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide500.pdf}
% \\
% $5.1s$
% &
% $5.2s$
% &
% $5.3s$
% \\
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide510.pdf}
% &
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide520.pdf}
% &
% \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide530.pdf}
% %\\
% %
% %\hline
% \end{tabular}
% %\caption{Distribution of sliding along the contact interface for Ricker wave
% %(gray scale given in meters)}
% \label{fig:3d_ricker1000_slide_9}
% \end{center}
% \end{figure}
% \end{tiny}
%
%
%
% \end{frame}
%
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{SPT Based Determination of Young's Modulus}
\begin{figure}[!hbpt]
\begin{center}
%
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01Ed.pdf}
\hfill
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_Histogram_Normal_01Ed.pdf}
%
\end{center}
\end{figure}
\vspace*{0.3cm}
Transformation of SPT $N$value $\rightarrow$ 1D Young's modulus, $E$ (cf. Phoon and Kulhawy (1999B))
Histogram of the residual (w.r.t the deterministic transformation equation) Young's modulus, along with fitted probability density function
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Uncertainty Propagation through Constitutive Eq.}
%
\begin{itemize}
\item Incremental elpl constitutive equation
$\displaystyle \Delta \sigma_{ij} = D_{ijkl} \displaystyle \Delta \epsilon_{kl}$
%\begin{normalsize}
%
% \begin{equation}
% \nonumber
% \frac{d\sigma_{ij}}{dt} = D_{ijkl} \frac{d\epsilon_{kl}}{dt}
% \end{equation}
\begin{eqnarray}
\nonumber
D_{ijkl} = \left\{\begin{array}{ll}
%
D^{el}_{ijkl}
%
%
\;\;\; & \mbox{\large{~for elastic}} \\
%
\\
%
D^{el}_{ijkl}

\frac{\displaystyle D^{el}_{ijmn} m_{mn} n_{pq} D^{el}_{pqkl}}
{\displaystyle n_{rs} D^{el}_{rstu} m_{tu}  \xi_* r_*}
\;\;\; & \mbox{\large{~for elasticplastic}}
%
\end{array} \right.
\end{eqnarray}
%\end{normalsize}
%\vspace{0.5cm}
% \item Nonlinear coupling in the ElPl modulus
\item What if all (any) material parameters are uncertain
\item PEP and SEPFEM methods for spatially variable and point uncertain material
% \item Focus on 1D $\rightarrow$ a nonlinear ODE with random coefficient and random forcing
%
%
%
% \begin{eqnarray}
% \nonumber
% \frac{d\sigma(x,t)}{dt} &=& \beta(\sigma(x,t),D^{el}(x),q(x),r(x);x,t) \frac{d\epsilon(x,t)}{dt} \\
% \nonumber
% &=& \eta(\sigma,D^{el},q,r,\epsilon; x,t) \mbox{\ \ \ \ with an I.C. $\sigma(0)=\sigma_0$}
% \end{eqnarray}
%
\end{itemize}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Probabilistic Stress Solution: \\ EulerianLagrangian form of FPK Equation}
%
% %
%
% %\begin{itemize}
% % 3D
% \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}
%
%
%
%
% % % 1D % 1D
% % 1D \begin{footnotesize}
% % 1D \begin{eqnarray}
% % 1D \nonumber
% % 1D &&\displaystyle \frac{\partial P(\sigma(x_t,t), t)}{\partial t}=
% % 1D  \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D^{el}(x_t),
% % 1D q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\
% % 1D \nonumber
% % 1D &+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t),
% % 1D \epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right. \\
% % 1D \nonumber
% % 1D & & \left. \left. \left. \eta(\sigma(x_{t\tau},t\tau), D^{el}(x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}),
% % 1D \epsilon(x_{t\tau},t\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} P(\sigma(x_t,t),t) \right] \\
% % 1D \nonumber
% % 1D &+& \displaystyle \frac{\partial^2}{\partial \sigma^2} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}}
% % 1D \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t), \epsilon(x_t,t)); \right. \right. \right. \\
% % 1D \nonumber
% % 1D & & \left. \left. \left. \eta (\sigma(x_{t\tau},t\tau), D^{el}(x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}),
% % 1D \epsilon(x_{t\tau},t\tau)) \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma (x_t,t),t) \right] \\
% % 1D \nonumber
% % 1D \end{eqnarray}
% % 1D
% % 1D \end{footnotesize}
%
%
%
%
%
% \end{frame}
%
%
%
%
%
%
%
% %
% % \item 6 equations
% %
% % \item Complete description of 3D probabilistic stressstrain response
% %
% % \end{itemize}
% %
% %
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{EulerianLagrangian FPK Equation and (SEP)FEM}
\begin{itemize}
\item Advectiondiffusion equation
%
\begin{equation}
\nonumber
\frac{\partial P(\sigma_{ij},t)}{\partial t}
=
\frac{\partial}{\partial\sigma_{ab}}
\left[N_{ab}^{(1)}P(\sigma_{ij},t)

\frac{\partial}{\partial \sigma_{cd}}
\left\{N_{abcd}^{(2)} P(\sigma_{ij},t)\right\} \right]
\end{equation}
%
\vspace*{0.1cm}
\item {\bf Complete} probabilistic description of response
\vspace*{0.1cm}
\item {\bf Secondorder exact} to covariance of time (exact mean and variance)
% 
%  \item Deterministic equation in probability density space
% 
%  \item Linear PDE in probability density space
%  $\rightarrow$ simplifies the numerical solution process
% 
%\item Applicable to any elasticplasticdamage material model (only coefficients $N_{ab}^{(1)}$
%and $N_{abcd}^{(2)}$ differ)
\vspace*{0.1cm}
\item Any uncertain FEM problem
(${\bf M} \ddot{\bf u}
+
{\bf C} \dot{\bf u}
+
{\bf K} {\bf u}
=
{\bf F}
$)
with
\begin{itemize}
\item uncertain material parameters (stiffness matrix ${\bf K}$),
\item uncertain loading (load vector ${\bf F}$)
\end{itemize}
can be analyzed using PEP and SEPFEM to obtain PDFs of DOFs,
stress, strain...
%  %\vspace*{0.2cm}
%  \item PEP solution is second order accurate (exact mean and standard deviation)
% 
%  %\vspace*{0.2cm}
%  \item SEPFEM solution (PDFs) can be made as accurate as need be
% 
% 
%  \item Tails of PDFs can than be used to develop accurate risk
% 
% 
%  \item Application to a realistic case of seismic wave propagation
%\vspace*{0.2truecm}
\end{itemize}
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Probabilistic ElasticPlastic Response}
%
%
% \begin{figure}[!hbpt]
% \begin{center}
% %\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDFedited.pdf}
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2012/DOELLNLworkshop2728Feb2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_Contour_PDFedited.pdf}
% \end{center}
% \end{figure}
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Probabilistic ElasticPlastic Response}
%
%
% \begin{figure}[!hbpt]
% \begin{center}
% %\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Conferences/2011/ICASP11_Zurich/Present/PDF_PlotEd.pdf}
% \includegraphics[width=9.5cm]{/home/jeremic/tex/works/Conferences/2012/DOELLNLworkshop2728Feb2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_PDFedited.pdf}
% \end{center}
% \end{figure}
%
% \end{frame}
%
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Spectral Stochastic ElasticPlastic FEM}
\begin{itemize}
\item Minimizing norm of error of finite representation using Galerkin
technique (Ghanem and Spanos 2003):
\vspace*{0.6truecm}
\begin{flushright}
\begin{equation}
\nonumber
\sum_{n = 1}^N K_{mn}^{ep} d_{ni} + \sum_{n = 1}^N \sum_{j = 0}^P d_{nj} \sum_{k = 1}^M C_{ijk} K_{mnk}^{'ep} = \left< F_m \psi_i[\{\xi_r\}] \right >
\end{equation}
\end{flushright}
% \begin{itemize}
%
% \vspace*{0.5cm}
% \item Final eqn.:
%
% \vspace*{0.4cm}
% \begin{flushright}
% \begin{normalsize}
% \begin{equation}
% \nonumber
% \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sum_{n = 1}^N K_{mn} d_{ni} + \sum_{n = 1}^N \sum_{j = 0}^P d_{nj} \sum_{k = 1}^M C_{ijk} K'_{mnk} = \left< F_m \psi_i[\{\zeta_r\}] \right >
% \end{equation}
% \end{normalsize}
% \end{flushright}
\vspace*{0.5cm}
\begin{equation}
\nonumber
K_{mn}^{ep} = \int_D B_n \textcolor{mycolor}{E}^{ep} B_m dV
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
K_{mnk}^{'ep} = \int_D B_n {\sqrt \lambda_k h_k} B_m dV
\end{equation}
\vspace*{1.0cm}
\begin{equation}
\nonumber
C_{ijk} = \left < \xi_k(\theta) \psi_i[\{\xi_r\}] \psi_j[\{\xi_r\}] \right >
\ \ \ \ \ \ \ \ \ \ \ \
F_m = \int_D \phi N_m dV \ \ \ \ \ \ \ \ \ \ \ \
\end{equation}
%\item FokkerPlanckKolmogorov approach based probabilistic constitutive integration
% at Gauss integration points
\end{itemize}
% \noindent Salient Features:
% \begin{itemize}
%
% \item Efficient representation of input random fields into finite number of random
% variables using KLexpansion
%
% \item Representation of (unknown) solution random variables using polynomial chaos of
% (known) input random variables
%
% \item FokkerPlanckKolmogorov approach based probabilistic constitutive integration
% at Gauss integration points
%
% \end{itemize}
%
%% \end{itemize}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Seismic Wave Propagation Through Uncertain Soils}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
% \frametitle{"Uniform" CPT Site Data}
%
%
% %\vspace*{0.7cm}
% %\begin{figure}
% \begin{center}
% \includegraphics[height=6.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/EastWestProfileEdited.pdf}
% \end{center}
% %\end{figure}
%
%
% \end{frame}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Full PDFs of all DOFs (and $\sigma_{ij}$, $\epsilon_{ij}$, etc.)}
%\frametitle{Full PDFs for Real Data Case}
\begin{itemize}
\vspace*{0.7cm}
\item Stochastic ElasticPlastic\\
Finite Element Method \\
(SEPFEM) \\
\vspace*{0.5cm}
\item Dynamic case
\vspace*{0.5cm}
\item Full PDF at \\
each time step $\Delta t$
\end{itemize}
\vspace*{4.60cm}
\begin{flushright}
\includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/EvolutionaryPDF_ActualEdited.pdf}
%\vspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{flushright}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{PDF at each $\Delta t$ (say at $6$ s)}
\begin{figure}
\begin{center}
\hspace*{1.75cm}
\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/PDFs_at6sec_Actual_vs_NoDataEdited.pdf}
\vspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{PDF $\rightarrow$ CDF (Fragility) at $6$ s}
\begin{figure}
\begin{center}
%\hspace*{0.75cm}
\includegraphics[width=8.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/CDFs_at6sec_Actual_vs_NoDataEdited.pdf}
\vspace*{0.75cm}
%\hspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
%
% \frametitle{Probability of Unacceptable Deformation ($50$cm)}
%
% \begin{figure}
% \begin{center}
% \vspace*{0.3cm}
% %\hspace*{0.75cm}
% \includegraphics[width=10.50cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/NewPlots/with_legends_and_labels/Exceedance50cm_LomaPrietaEdited_ps.pdf}
% \vspace*{0.5cm}
% %\hspace*{0.75cm}
% %\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
% \end{center}
% \end{figure}
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \section{Probabilistic Modeling}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \subsection{Uncertain (Geo) Materials}
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM \begin{frame}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \frametitle{Uncertainty Propagation through Constitutive Eq.}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \begin{itemize}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \item Incremental elpl constitutive equation
% MOZDA PEP SEPFEM $\displaystyle \Delta \sigma_{ij} = D_{ijkl} \displaystyle \Delta \epsilon_{kl}$
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %\begin{normalsize}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \begin{equation}
% MOZDA PEP SEPFEM % \nonumber
% MOZDA PEP SEPFEM % \frac{d\sigma_{ij}}{dt} = D_{ijkl} \frac{d\epsilon_{kl}}{dt}
% MOZDA PEP SEPFEM % \end{equation}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \begin{eqnarray}
% MOZDA PEP SEPFEM \nonumber
% MOZDA PEP SEPFEM D_{ijkl} = \left\{\begin{array}{ll}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM D^{el}_{ijkl}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM \;\;\; & \mbox{\large{~for elastic}} \\
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM \\
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM D^{el}_{ijkl}
% MOZDA PEP SEPFEM 
% MOZDA PEP SEPFEM \frac{\displaystyle D^{el}_{ijmn} m_{mn} n_{pq} D^{el}_{pqkl}}
% MOZDA PEP SEPFEM {\displaystyle n_{rs} D^{el}_{rstu} m_{tu}  \xi_* r_*}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \;\;\; & \mbox{\large{~for elasticplastic}}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM \end{array} \right.
% MOZDA PEP SEPFEM \end{eqnarray}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %\end{normalsize}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %\vspace{0.5cm}
% MOZDA PEP SEPFEM % \item Nonlinear coupling in the ElPl modulus
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \item What if all (any) material parameters are uncertain
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \item PEP and SEPFEM methods for spatially variable and point uncertain material
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM % \item Focus on 1D $\rightarrow$ a nonlinear ODE with random coefficient and random forcing
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \begin{eqnarray}
% MOZDA PEP SEPFEM % \nonumber
% MOZDA PEP SEPFEM % \frac{d\sigma(x,t)}{dt} &=& \beta(\sigma(x,t),D^{el}(x),q(x),r(x);x,t) \frac{d\epsilon(x,t)}{dt} \\
% MOZDA PEP SEPFEM % \nonumber
% MOZDA PEP SEPFEM % &=& \eta(\sigma,D^{el},q,r,\epsilon; x,t) \mbox{\ \ \ \ with an I.C. $\sigma(0)=\sigma_0$}
% MOZDA PEP SEPFEM % \end{eqnarray}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \end{itemize}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \end{frame}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM \begin{frame}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \frametitle{Probabilistic Stress Solution: \\ EulerianLagrangian form of FPK Equation}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %\begin{itemize}
% MOZDA PEP SEPFEM % 3D
% MOZDA PEP SEPFEM \begin{footnotesize}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \begin{eqnarray}
% MOZDA PEP SEPFEM \nonumber
% MOZDA PEP SEPFEM \lefteqn{\displaystyle \frac{\partial P(\sigma_{ij}(x_t,t), t)}{\partial t} = \displaystyle \frac{\partial}{\partial \sigma_{mn}}
% MOZDA PEP SEPFEM \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.} \\
% MOZDA PEP SEPFEM \nonumber
% MOZDA PEP SEPFEM &+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[\displaystyle \frac{\partial
% MOZDA PEP SEPFEM \eta_{mn}(\sigma_{mn}(x_t,t), E_{mnrs}(x_t),
% MOZDA PEP SEPFEM \epsilon_{rs}(x_t,t))} {\partial \sigma_{ab}}; \right. \right. \right. \\
% MOZDA PEP SEPFEM \nonumber
% MOZDA PEP SEPFEM & & \left. \left. \left. \eta_{ab} (\sigma_{ab}(x_{t\tau}, t\tau), E_{abcd}(x_{t\tau}), \epsilon_{cd}(x_{t\tau}, t\tau)
% MOZDA PEP SEPFEM \vphantom{\int_{0}^{t}} \right] \right \} P(\sigma_{ij}(x_t,t),t) \right] \\
% MOZDA PEP SEPFEM \nonumber
% MOZDA PEP SEPFEM &+& \displaystyle \frac{\partial^2}{\partial \sigma_{mn} \partial \sigma_{ab}} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[
% MOZDA PEP SEPFEM \vphantom{\int_{0}^{t}} \eta_{mn}(\sigma_{mn}(x_t,t), E_{mnrs}(x_t), \epsilon_{rs}(x_t,t)); \right. \right. \right. \\
% MOZDA PEP SEPFEM \nonumber
% MOZDA PEP SEPFEM & & \left. \left. \left. \eta_{ab} (\sigma_{ab}(x_{t\tau}, t\tau), E_{abcd}(x_{t\tau}), \epsilon_{cd}(x_{t\tau}, t\tau))
% MOZDA PEP SEPFEM \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma_{ij}(x_t,t),t) \right]
% MOZDA PEP SEPFEM \end{eqnarray}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \end{footnotesize}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM % 1D % 1D
% MOZDA PEP SEPFEM % 1D \begin{footnotesize}
% MOZDA PEP SEPFEM % 1D \begin{eqnarray}
% MOZDA PEP SEPFEM % 1D \nonumber
% MOZDA PEP SEPFEM % 1D &&\displaystyle \frac{\partial P(\sigma(x_t,t), t)}{\partial t}=
% MOZDA PEP SEPFEM % 1D  \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D^{el}(x_t),
% MOZDA PEP SEPFEM % 1D q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\
% MOZDA PEP SEPFEM % 1D \nonumber
% MOZDA PEP SEPFEM % 1D &+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t),
% MOZDA PEP SEPFEM % 1D \epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right. \\
% MOZDA PEP SEPFEM % 1D \nonumber
% MOZDA PEP SEPFEM % 1D & & \left. \left. \left. \eta(\sigma(x_{t\tau},t\tau), D^{el}(x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}),
% MOZDA PEP SEPFEM % 1D \epsilon(x_{t\tau},t\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} P(\sigma(x_t,t),t) \right] \\
% MOZDA PEP SEPFEM % 1D \nonumber
% MOZDA PEP SEPFEM % 1D &+& \displaystyle \frac{\partial^2}{\partial \sigma^2} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}}
% MOZDA PEP SEPFEM % 1D \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t), \epsilon(x_t,t)); \right. \right. \right. \\
% MOZDA PEP SEPFEM % 1D \nonumber
% MOZDA PEP SEPFEM % 1D & & \left. \left. \left. \eta (\sigma(x_{t\tau},t\tau), D^{el}(x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}),
% MOZDA PEP SEPFEM % 1D \epsilon(x_{t\tau},t\tau)) \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma (x_t,t),t) \right] \\
% MOZDA PEP SEPFEM % 1D \nonumber
% MOZDA PEP SEPFEM % 1D \end{eqnarray}
% MOZDA PEP SEPFEM % 1D
% MOZDA PEP SEPFEM % 1D \end{footnotesize}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \end{frame}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \item 6 equations
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \item Complete description of 3D probabilistic stressstrain response
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \end{itemize}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM \begin{frame}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \frametitle{EulerianLagrangian FPK Equation and (SEP)FEM}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \begin{itemize}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \item Advectiondiffusion equation
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM \begin{equation}
% MOZDA PEP SEPFEM \nonumber
% MOZDA PEP SEPFEM \frac{\partial P(\sigma_{ij},t)}{\partial t}
% MOZDA PEP SEPFEM =
% MOZDA PEP SEPFEM \frac{\partial}{\partial\sigma_{ab}}
% MOZDA PEP SEPFEM \left[N_{ab}^{(1)}P(\sigma_{ij},t)
% MOZDA PEP SEPFEM 
% MOZDA PEP SEPFEM \frac{\partial}{\partial \sigma_{cd}}
% MOZDA PEP SEPFEM \left\{N_{abcd}^{(2)} P(\sigma_{ij},t)\right\} \right]
% MOZDA PEP SEPFEM \end{equation}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \vspace*{0.1cm}
% MOZDA PEP SEPFEM \item {\bf Complete} probabilistic description of response
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \vspace*{0.1cm}
% MOZDA PEP SEPFEM \item {\bf Secondorder exact} to covariance of time (exact mean and variance)
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM % 
% MOZDA PEP SEPFEM %  \item Deterministic equation in probability density space
% MOZDA PEP SEPFEM % 
% MOZDA PEP SEPFEM %  \item Linear PDE in probability density space
% MOZDA PEP SEPFEM %  $\rightarrow$ simplifies the numerical solution process
% MOZDA PEP SEPFEM % 
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %\item Applicable to any elasticplasticdamage material model (only coefficients $N_{ab}^{(1)}$
% MOZDA PEP SEPFEM %and $N_{abcd}^{(2)}$ differ)
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \vspace*{0.1cm}
% MOZDA PEP SEPFEM \item Any uncertain FEM problem
% MOZDA PEP SEPFEM (${\bf M} \ddot{\bf u}
% MOZDA PEP SEPFEM +
% MOZDA PEP SEPFEM {\bf C} \dot{\bf u}
% MOZDA PEP SEPFEM +
% MOZDA PEP SEPFEM {\bf K} {\bf u}
% MOZDA PEP SEPFEM =
% MOZDA PEP SEPFEM {\bf F}
% MOZDA PEP SEPFEM $)
% MOZDA PEP SEPFEM with
% MOZDA PEP SEPFEM \begin{itemize}
% MOZDA PEP SEPFEM \item uncertain material parameters (stiffness matrix ${\bf K}$),
% MOZDA PEP SEPFEM \item uncertain loading (load vector ${\bf F}$)
% MOZDA PEP SEPFEM \end{itemize}
% MOZDA PEP SEPFEM can be analyzed using PEP and SEPFEM to obtain PDFs of DOFs,
% MOZDA PEP SEPFEM stress, strain...
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %  %\vspace*{0.2cm}
% MOZDA PEP SEPFEM %  \item PEP solution is second order accurate (exact mean and standard deviation)
% MOZDA PEP SEPFEM % 
% MOZDA PEP SEPFEM %  %\vspace*{0.2cm}
% MOZDA PEP SEPFEM %  \item SEPFEM solution (PDFs) can be made as accurate as need be
% MOZDA PEP SEPFEM % 
% MOZDA PEP SEPFEM % 
% MOZDA PEP SEPFEM %  \item Tails of PDFs can than be used to develop accurate risk
% MOZDA PEP SEPFEM % 
% MOZDA PEP SEPFEM % 
% MOZDA PEP SEPFEM %  \item Application to a realistic case of seismic wave propagation
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %\vspace*{0.2truecm}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \end{itemize}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \end{frame}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM \begin{frame}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \frametitle{Probabilistic ElasticPlastic Response}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \begin{figure}[!hbpt]
% MOZDA PEP SEPFEM \begin{center}
% MOZDA PEP SEPFEM %\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDFedited.pdf}
% MOZDA PEP SEPFEM \includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2012/DOELLNLworkshop2728Feb2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_Contour_PDFedited.pdf}
% MOZDA PEP SEPFEM \end{center}
% MOZDA PEP SEPFEM \end{figure}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \end{frame}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM \begin{frame}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \frametitle{Probabilistic ElasticPlastic Response}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \begin{figure}[!hbpt]
% MOZDA PEP SEPFEM \begin{center}
% MOZDA PEP SEPFEM %\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Conferences/2011/ICASP11_Zurich/Present/PDF_PlotEd.pdf}
% MOZDA PEP SEPFEM \includegraphics[width=9.5cm]{/home/jeremic/tex/works/Conferences/2012/DOELLNLworkshop2728Feb2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_PDFedited.pdf}
% MOZDA PEP SEPFEM \end{center}
% MOZDA PEP SEPFEM \end{figure}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \end{frame}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \begin{frame}
% MOZDA PEP SEPFEM \frametitle{Spectral Stochastic ElasticPlastic FEM}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \begin{itemize}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \item Minimizing norm of error of finite representation using Galerkin
% MOZDA PEP SEPFEM technique (Ghanem and Spanos 2003):
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \vspace*{0.6truecm}
% MOZDA PEP SEPFEM \begin{flushright}
% MOZDA PEP SEPFEM \begin{equation}
% MOZDA PEP SEPFEM \nonumber
% MOZDA PEP SEPFEM \sum_{n = 1}^N K_{mn}^{ep} d_{ni} + \sum_{n = 1}^N \sum_{j = 0}^P d_{nj} \sum_{k = 1}^M C_{ijk} K_{mnk}^{'ep} = \left< F_m \psi_i[\{\xi_r\}] \right >
% MOZDA PEP SEPFEM \end{equation}
% MOZDA PEP SEPFEM \end{flushright}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM % \begin{itemize}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \vspace*{0.5cm}
% MOZDA PEP SEPFEM % \item Final eqn.:
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \vspace*{0.4cm}
% MOZDA PEP SEPFEM % \begin{flushright}
% MOZDA PEP SEPFEM % \begin{normalsize}
% MOZDA PEP SEPFEM % \begin{equation}
% MOZDA PEP SEPFEM % \nonumber
% MOZDA PEP SEPFEM % \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sum_{n = 1}^N K_{mn} d_{ni} + \sum_{n = 1}^N \sum_{j = 0}^P d_{nj} \sum_{k = 1}^M C_{ijk} K'_{mnk} = \left< F_m \psi_i[\{\zeta_r\}] \right >
% MOZDA PEP SEPFEM % \end{equation}
% MOZDA PEP SEPFEM % \end{normalsize}
% MOZDA PEP SEPFEM % \end{flushright}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \vspace*{0.5cm}
% MOZDA PEP SEPFEM \begin{equation}
% MOZDA PEP SEPFEM \nonumber
% MOZDA PEP SEPFEM K_{mn}^{ep} = \int_D B_n \textcolor{mycolor}{E}^{ep} B_m dV
% MOZDA PEP SEPFEM \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
% MOZDA PEP SEPFEM K_{mnk}^{'ep} = \int_D B_n {\sqrt \lambda_k h_k} B_m dV
% MOZDA PEP SEPFEM \end{equation}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \vspace*{1.0cm}
% MOZDA PEP SEPFEM \begin{equation}
% MOZDA PEP SEPFEM \nonumber
% MOZDA PEP SEPFEM C_{ijk} = \left < \xi_k(\theta) \psi_i[\{\xi_r\}] \psi_j[\{\xi_r\}] \right >
% MOZDA PEP SEPFEM \ \ \ \ \ \ \ \ \ \ \ \
% MOZDA PEP SEPFEM F_m = \int_D \phi N_m dV \ \ \ \ \ \ \ \ \ \ \ \
% MOZDA PEP SEPFEM \end{equation}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %\item FokkerPlanckKolmogorov approach based probabilistic constitutive integration
% MOZDA PEP SEPFEM % at Gauss integration points
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \end{itemize}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM % \noindent Salient Features:
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM % \begin{itemize}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \item Efficient representation of input random fields into finite number of random
% MOZDA PEP SEPFEM % variables using KLexpansion
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \item Representation of (unknown) solution random variables using polynomial chaos of
% MOZDA PEP SEPFEM % (known) input random variables
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \item FokkerPlanckKolmogorov approach based probabilistic constitutive integration
% MOZDA PEP SEPFEM % at Gauss integration points
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \end{itemize}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM %% \end{itemize}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM \end{frame}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %\subsection{Seismic Wave Propagation Through Uncertain Soils}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \begin{frame}
% MOZDA PEP SEPFEM % \frametitle{"Uniform" CPT Site Data}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % %\vspace*{0.7cm}
% MOZDA PEP SEPFEM % %\begin{figure}
% MOZDA PEP SEPFEM % \begin{center}
% MOZDA PEP SEPFEM % \includegraphics[height=6.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/EastWestProfileEdited.pdf}
% MOZDA PEP SEPFEM % \end{center}
% MOZDA PEP SEPFEM % %\end{figure}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \end{frame}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \begin{frame}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \frametitle{Full PDFs of all DOFs (and $\sigma_{ij}$, $\epsilon_{ij}$, etc.)}
% MOZDA PEP SEPFEM %\frametitle{Full PDFs for Real Data Case}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \begin{itemize}
% MOZDA PEP SEPFEM \vspace*{0.7cm}
% MOZDA PEP SEPFEM \item Stochastic ElasticPlastic\\
% MOZDA PEP SEPFEM Finite Element Method \\
% MOZDA PEP SEPFEM (SEPFEM) \\
% MOZDA PEP SEPFEM \vspace*{0.5cm}
% MOZDA PEP SEPFEM \item Dynamic case
% MOZDA PEP SEPFEM \vspace*{0.5cm}
% MOZDA PEP SEPFEM \item Full PDF at \\
% MOZDA PEP SEPFEM each time step $\Delta t$
% MOZDA PEP SEPFEM \end{itemize}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM \vspace*{4.60cm}
% MOZDA PEP SEPFEM \begin{flushright}
% MOZDA PEP SEPFEM \includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/EvolutionaryPDF_ActualEdited.pdf}
% MOZDA PEP SEPFEM %\vspace*{0.75cm}
% MOZDA PEP SEPFEM %\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
% MOZDA PEP SEPFEM \end{flushright}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM \end{frame}
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \begin{frame}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \frametitle{PDF at each $\Delta t$ (say at $6$ s)}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \begin{figure}
% MOZDA PEP SEPFEM % \begin{center}
% MOZDA PEP SEPFEM % \hspace*{1.75cm}
% MOZDA PEP SEPFEM % \includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/PDFs_at6sec_Actual_vs_NoDataEdited.pdf}
% MOZDA PEP SEPFEM % \vspace*{0.75cm}
% MOZDA PEP SEPFEM % %\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
% MOZDA PEP SEPFEM % \end{center}
% MOZDA PEP SEPFEM % \end{figure}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % %
% MOZDA PEP SEPFEM % \end{frame}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \begin{frame}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \frametitle{PDF $\rightarrow$ CDF (Fragility) at $6$ s}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \begin{figure}
% MOZDA PEP SEPFEM % \begin{center}
% MOZDA PEP SEPFEM % %\hspace*{0.75cm}
% MOZDA PEP SEPFEM % \includegraphics[width=8.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/CDFs_at6sec_Actual_vs_NoDataEdited.pdf}
% MOZDA PEP SEPFEM % \vspace*{0.75cm}
% MOZDA PEP SEPFEM % %\hspace*{0.75cm}
% MOZDA PEP SEPFEM % %\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
% MOZDA PEP SEPFEM % \end{center}
% MOZDA PEP SEPFEM % \end{figure}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % %
% MOZDA PEP SEPFEM % \end{frame}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \begin{frame}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \frametitle{Probability of Unacceptable Deformation ($50$cm)}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \begin{figure}
% MOZDA PEP SEPFEM % \begin{center}
% MOZDA PEP SEPFEM % \vspace*{0.3cm}
% MOZDA PEP SEPFEM % %\hspace*{0.75cm}
% MOZDA PEP SEPFEM % \includegraphics[width=10.50cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/NewPlots/with_legends_and_labels/Exceedance50cm_LomaPrietaEdited_ps.pdf}
% MOZDA PEP SEPFEM % \vspace*{0.5cm}
% MOZDA PEP SEPFEM % %\hspace*{0.75cm}
% MOZDA PEP SEPFEM % %\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
% MOZDA PEP SEPFEM % \end{center}
% MOZDA PEP SEPFEM % \end{figure}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % \end{frame}
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM %
% MOZDA PEP SEPFEM % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MOZDA PEP SEPFEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Summary}
\begin{itemize}
\item High fidelity, extensive V \& V, time domain, nonlinear, earthquake soil structure
interaction (ESSI) modeling and simulations (deterministic and probabilistic)
for nuclear power plant licensing and design
\vspace*{0.2cm}
\item The ESSI Simulator System (Program, Computer, Notes)
\vspace*{0.2cm}
\item Educational effort is essential (USNRC, CNSC, IAEA, LBNL, INL,
companies), seminars, short courses
%\vspace*{0.1cm}
%\item Information Portal:\\
%{\large \bf
%\url{http://nrcessisimulator.info}}
\vspace*{0.2cm}
\item Funding from the USNRC, USDOE, USNSF, and the CNSC is much appreciated
\end{itemize}
\end{frame}
\end{document}