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%\title{NRC Staff Capacity Building: \\
% Micromechanical Origins of ElastoPlasticity }
\title{Aspects of Deterministic and Probabilistic \\
Modeling and Simulation in Earthquake Engineering }
%\subtitle
%{Include Only If Paper Has a Subtitle}
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\author[Jeremi{\'c}] % (optional, use only with lots of authors)
{Boris~Jeremi{\'c} \\
{\small with
Dr. Tafazzoli,
Mr. Abell,
Mr. Jeong,
Dr. Pisan{\`o},
Prof. Sett (UA),
Prof. Taiebat (UBC),
Prof. Yang (UAA),
Dr. Cheng (Itasca)}
}
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%{ Professor, University of California, Davis\\
{ University of California, Davis\\
% and\\
% Faculty Scientist, Lawrence Berkeley National Laboratory, Berkeley }
Lawrence Berkeley National Laboratory, Berkeley }
%  Use the \inst command only if there are several affiliations.
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{\small LBNL Seminar, December 2012}
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\section{Introduction}
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\subsection{Motivation}
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\begin{frame}
\frametitle{The Problem}
\begin{itemize}
\item Seismic response of Nuclear Power Plants
\vspace*{0.1cm}
\item 3D, inclined seismic motions consisting of body and surface waves
\vspace*{0.1cm}
\item Inelastic (elastic, damage, plastic behavior of materials: soil, rock,
concrete, steel, rubber, etc.)
\vspace*{0.1cm}
\item Full coupling of pore fluids (in soil and rock) with soil/rock skeleton
\vspace*{0.1cm}
\item Buoyant effects (foundations below water table)
\vspace*{0.1cm}
\item Uncertainty in seismic sources, path, soil/rock response and structural
response
\end{itemize}
\end{frame}
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\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
\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}
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\subsection{NRC ESSI Simulator System}
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%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame}
%  \frametitle{Project Goals}
% 
%  \begin{itemize}
% 
%  %\vspace*{0.5cm}
%  \item Development of the NRC ESSI Simulator System
%  for HiFi modeling and simulation of nonlinear earthquake
%  soil/rock structure interaction problems:
%  \begin{itemize}
%  % \item Time domain, nonlinear, parallel finite element program:
%  \item {\bf NRCESSIProgram}
%  % \item High performance, parallel computer:
%  \item {\bf NRCESSIComputer}
%  % \item Educational endeavor, documentation:
%  \item {\bf NRCESSINotes}
%  \end{itemize}
% 
%  \vspace*{0.1cm}
%  \item Education: NRC Staff Capacity Building (seminars, short courses,
%  NRC ESSI Notes, advising), targeting wider audience as well
% 
%  \vspace*{0.1cm}
%  \item Development of ESSI case studies:
%  3D, inclined seismic motions, soil/rock;
%  foundation interface slip, seismic energy propagation dissipation
% 
%  %\vspace*{0.2cm}
%  %\item
%  %
%  %\vspace*{0.2cm}
%  %\item
%  %
%  %\vspace*{0.2cm}
%  %\item
% 
%  \end{itemize}
% 
%  \end{frame}
% 
% 
% 
% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{NRC ESSI Simulator System}
\begin{itemize}
\item {\bf The NRCESSIProgram} 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 NRCESSIComputer. \
%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 NRCESSIComputer} 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 NRCESSINotes} 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}
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\begin{frame}
\frametitle{NRC 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 Various public domain licenses (GPL, LGPL, BSD, CC)
%\vspace*{0.3cm}
\vspace*{0.1cm}
\item Verification and Validation
\vspace*{0.1cm}
\item Detailed program documentation (part of NRC ESSI Notes)
\vspace*{0.1cm}
\item Target users: U.S.NRC staff, UCD students, external users
%\item Sources will be available through
%{\bf
%\url{http://nrcessisimulator.info}}
\end{itemize}
\end{frame}
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\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}
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\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{NRC 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 supercomputers
%\vspace*{0.1cm}
\item Current system: 208 CPUs
%\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}
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\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{NRC ESSI Simulator 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}
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\begin{frame}
\frametitle{NRC 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}
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\section{Deterministic Modeling}
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\subsection{Seismic Energy Flow, Finite Elements, Material Models, Loading, HPC}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
% \frametitle{Seismic Energy Dissipation for \underline{Soil}FoundationStructure Systems}
\frametitle{High Fidelity Modeling}
% \frametitle{Seismic Energy Dissipation for
% \underline{Soil}FoundationStructure Systems}
\begin{itemize}
\item 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 SFS domain:
\begin{itemize}
\item {Radiation} of reflected waves
\item {Radiation} of oscillating SFS system
\end{itemize}
\vspace*{0.1cm}
\item Mechanical dissipation inside SFS 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 interface
\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}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{High Performance, Parallel Computing}
\begin{itemize}
\item The NRC ESSI Simulator 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
\vspace*{0.2cm}
\item Developed for multiple/variable capability CPUs and
networks (DMP and
SMPs)
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame}
%  \frametitle{Earthquake Ground Motions}
% 
% 
% 
%  \begin{itemize}
% 
% 
%  %\vspace*{0.5cm}
%  \item Realistic earthquake ground motions
%  \begin{itemize}
%  \item Body: P and S waves
%  \item Surface: Rayleigh, Love waves, etc.
%  \item Lack of correlation (incoherence)
%  \item Inclined waves
%  \item 3D waves
%  \item Earthquake energy dissipation
% 
%  \end{itemize}
% 
% 
% 
% 
% 
%  \end{itemize}
% 
% 
% 
% 
%  \end{frame}
% 
% 
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% 
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%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \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*{0.5cm}
%  \\
%  \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}
% 
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\begin{frame}
\frametitle{Representative NPP Example Problem}
\vspace*{0.50cm}
\begin{itemize}
\item Body and surface seismic waves
\item Seismic wave frequencies \\
up to $50$Hz
\item Elasticplastic soil/rock and \\
structural components,
\item Inelastic contact/gap
\item Seismic isolator effects
\item Buoyant effects for deep foundation embedment
\item High Fidelity Model: soil block: $230m \times 230m \times100m$, foundation
$90m \times 90m$ Containment Structure: $40m \times 50m$, 2.1 Million DOFs,
700,000 elements,
\end{itemize}
\vspace*{6.0cm}
\begin{flushright}
\hspace*{1cm}
\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Conferences/2012/Seismic_research_issues_for_NPPs/Large_NPP_model_01A.jpg}
\end{flushright}
\end{frame}
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\begin{frame}
\frametitle{Finite Elements}
\begin{itemize}
\item Linear and nonlinear truss element
\item Linear and nonlinear beam (disp. based), variable BCs
\item Linear shell Triangle and Quad with drilling DOFs
\item Linear and nonlinear thick
shell
\item Single phase solid bricks (8, 20, 820, 27 node element)
\item Two phase (fully coupled, porous solid, pore fluid) solid bricks (8 and
27 node: $upU$, $up$)
\item Dry friction slip and gap element
\item Saturated gap and (effective stress) slip element
\item Seismic isolator (latex rubber, neoprene rubber, rubber with lead core,
friction pendulum)
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Material Models for Solids and Structures}
\begin{itemize}
\item Elastic: linear, nonlinear isotropic, cross anisotropic
\vspace*{0.3cm}
\item ElasticPlastic: von Mises, DruckerPrager,
CamClay, Rounded MohrCoulomb, Parabolic Leon,
SANIsand (DafaliasManzari...), Pisan{\`o}Jeremi{\'c}.
\vspace*{0.3cm}
\item Isotropic and kinematic
(translational and rotational)
kinematic hardening
\end{itemize}
%\vspace*{2.0cm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{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
%\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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Verification and Validation Suite}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Verification, Validation 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}
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\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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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*{1.0truecm}
\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*{0.3cm}
\item Material modeling and simulation (elastic, elasticplastic...)
\vspace*{0.3cm}
\item Finite elements (solids, structural, special...)
\vspace*{0.3cm}
\item Solution advancement algorithms (static, dynamic...)
\vspace*{0.3cm}
\item Seismic input and radiation
\vspace*{0.3cm}
\item Finite element model verification
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Mesh Size Effects on Seismic Wave Propagation Modeling}
\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{3D Inclined Body and Surface Seismic Wave Fields}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Free Field, Inclined, 3D Body and Surface Waves}
\begin{itemize}
\item Development of analytic and numerical 3D, inclined, uncorrelated
seismic motions for verification
\vspace*{0.2cm}
\item Large scale models
\vspace*{0.2cm}
\item Point shear source
\vspace*{0.2cm}
\item Stress drop:
\begin{itemize}
\item Wavelet (Ricker, \\
Ormsby, etc)
\item Analytic
\end{itemize}
\vspace*{0.2cm}
\item Seismic input using DRM
\end{itemize}
\vspace*{4.6cm}
\begin{flushright}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/FaultSlipModel2km.pdf}
\end{flushright}
\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}
%
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% Ex out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Ex out \subsection{Examples}
% Ex out %
% Ex out
% Ex out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Ex out \begin{frame}
% Ex out \frametitle{Few Illustrative Examples}
% Ex out
% Ex out \begin{itemize}
% Ex out \item Slip between foundation slab and the soil/rock
% Ex out underneath
% Ex out
% Ex out \vspace*{0.2cm}
% Ex out \item Passive seismic isolation by liquefaction
% Ex out
% Ex out \vspace*{0.2cm}
% Ex out \item Structural response in liquefied soil
% Ex out
% Ex out
% Ex out \end{itemize}
% Ex out
% Ex out \end{frame}
% Ex out
% Ex out
% Ex out
% Ex out
% Ex out %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Ex out \begin{frame}
% Ex out \frametitle{Nuclear Power Plant with Base Slip}
% Ex out
% Ex out \begin{itemize}
% Ex out
% Ex out \item Low friction zone between \\
% Ex out concrete foundation and soil/rock
% Ex out
% Ex out \item Inclined, 3D, body and surface, \\
% Ex out seismic wave field (wavelets: \\
% Ex out Ricker, Ormsby; real seismic, etc.)
% Ex out
% Ex out
% Ex out \end{itemize}
% Ex out
% Ex out
% Ex out
% Ex out \vspace*{4.0cm}
% Ex out \begin{figure}[!h]
% Ex out \begin{flushright}
% Ex out \includegraphics[width=2.50cm]{/home/jeremic/tex/works/Conferences/2011/NRC_LBNL_Review_Panel_Sept2011/2D_faul_slip_model_top_200m.pdf}
% Ex out %{\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Conferences/2011/NRC_LBNL_Review_Panel_Feb2011/Case_study_model/visit0002.jpeg}}
% Ex out \end{flushright}
% Ex out \end{figure}
% Ex out
% Ex out \vspace*{0.9cm}
% Ex out \begin{figure}[!h]
% Ex out \begin{flushright}
% Ex out {\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Conferences/2011/NRC_LBNL_Review_Panel_Feb2011/Case_study_model/visit0002.jpeg}}
% Ex out \end{flushright}
% Ex out \end{figure}
% Ex out
% Ex out
% Ex out
% Ex out
% Ex out \vspace*{3.6cm}
% Ex out \begin{figure}[H]
% Ex out \begin{center}
% Ex out %\vspace*{0.5cm}
% Ex out %\includegraphics[width=2.0cm]{/home/jeremic/tex/works/Conferences/2011/NRC_LBNL_Review_Panel_Sept2011/2D_faul_slip_model_top_200m.pdf}
% Ex out %\hspace*{0.5cm}
% Ex out %\vspace*{0.2cm}
% Ex out \includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/FreeFieldInclinedMotionModels/FaultSlip_Ormsby/figs/surface_200m/middle_acceleration_x.pdf}
% Ex out \hspace*{0.5cm}
% Ex out \includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/FreeFieldInclinedMotionModels/FaultSlip_Ormsby/figs/surface_200m/middle_acceleration_z.pdf}
% Ex out %
% Ex out \vspace*{0.5cm}
% Ex out \hspace*{0.8cm}
% Ex out \mbox{horizontal}
% Ex out \hspace*{4cm}
% Ex out \mbox{vertical}
% Ex out \hspace*{3cm}
% Ex out \end{center}
% Ex out \end{figure}
% Ex out \vspace*{1.0cm}
% Ex out %
% Ex out % %\vspace*{3.5cm}
% Ex out % \begin{figure}[H]
% Ex out % \begin{center}
% Ex out % \includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/FreeFieldInclinedMotionModels/FaultSlip_Ormsby/figs/surface_200m/middle_acceleration_x.pdf}
% Ex out % \hspace*{0.5cm}
% Ex out % \includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/FreeFieldInclinedMotionModels/FaultSlip_Ormsby/figs/surface_200m/middle_acceleration_z.pdf}
% Ex out % \end{center}
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% Ex out % {horizontal accelerations \hfill vertical accelerations}
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% Ex out \frametitle{Acc. Response for a Full 3D (at $45^\circ$) Ricker Wavelet}
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% Ex out \begin{figure}[H]
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% Ex out %\hline
% Ex out \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}
% Ex out &
% Ex out \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}
% Ex out \\
% Ex out \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}
% Ex out &
% Ex out \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}
% Ex out \end{tabular}
% Ex out %\caption{Comparison of acceleration time histories of the structure between
% Ex out %slipping and noslipping models for Ricker wave}
% Ex out \label{fig:3d_ricker_acc_1000}
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% Ex out \frametitle{FFT Response for a Full 3D (at $45^\circ$) Ricker Wavelet}
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% Ex out \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}
% Ex out &
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% Ex out \\
% Ex out \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}
% Ex out &
% Ex out \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}
% Ex out \end{tabular}
% Ex out %\caption{Comparison of FFT of the acceleration of the structure between
% Ex out %slipping and noslipping models for Ricker wave}
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% Ex out \frametitle{Gaping Response ($45^\circ$ Ricker Wavelet)}
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% Ex out \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap450.pdf}
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% Ex out \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap480.pdf}
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% Ex out \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap510.pdf}
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% Ex out \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap520.pdf}
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% Ex out %\\
% Ex out %
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% Ex out \end{tabular}
% Ex out %\caption{Distribution of gap openings along the contact interface for Ricker wave
% Ex out %(gray scale given in meters)}
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% Ex out \frametitle{Slipping Response and Energy Dissipated ($45^\circ$ Ricker)}
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% Ex out %\hline
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% Ex out \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide450.pdf}
% Ex out &
% Ex out \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide460.pdf}
% Ex out &
% Ex out \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide470.pdf}
% Ex out \\
% Ex out $4.8s$
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% Ex out \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide480.pdf}
% Ex out &
% Ex out \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide490.pdf}
% Ex out &
% Ex out \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide500.pdf}
% Ex out \\
% Ex out $5.1s$
% Ex out &
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% Ex out \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide510.pdf}
% Ex out &
% Ex out \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide520.pdf}
% Ex out &
% Ex out \includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide530.pdf}
% Ex out %\\
% Ex out %
% Ex out %\hline
% Ex out \end{tabular}
% Ex out %\caption{Distribution of sliding along the contact interface for Ricker wave
% Ex out %(gray scale given in meters)}
% Ex out \label{fig:3d_ricker1000_slide_9}
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% Ex out \includegraphics[width=4cm]{/home/jeremic/tex/works/Conferences/2009/CompDyn/Present/StackElementsCompare.pdf}
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% Ex out %\includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/MomentBent1Pile2.pdf}
% Ex out %\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
% Ex out %bridge.}
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\section{Probabilistic Modeling}
\subsection{Uncertain (Geo) Materials}
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\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 Spatial \\
variability
\vspace*{0.5cm}
\item Pointwise \\
uncertainty, \\
testing \\
error, \\
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}
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%  \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}
% 
% 
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\begin{frame}
\frametitle{Types of Uncertainties}
\begin{itemize}
\item Aleatory uncertainty  inherent variation of physical system
%
\begin{itemize}
%
\item Can not be reduced
%
\item Has highly developed mathematical tools
%
\end{itemize}
%
\vspace*{0.2cm}
\item Epistemic uncertainty  due to lack of knowledge
\begin{itemize}
\item Can be reduced by \\
collecting more data
\item Mathematical tools \\
are not well developed
\item tradeoff with \\
aleatory uncertainty
\end{itemize}
%
\vspace*{3.2cm}
\begin{figure}[!hbpt]
\begin{flushright}
\includegraphics[height=5cm,angle=90]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/uncertain03.pdf}
\end{flushright}
\end{figure}
%\vspace*{1.0cm}
\item Ergodicity (exchanging ensemble averages for time average) assumed to hold
\end{itemize}
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\begin{frame}
\frametitle{Recent StateoftheArt}
\begin{itemize}
%\vspace*{0.5cm}
\item Governing equation
% \vspace*{0.5cm}
\begin{itemize}
\item Dynamic problems $\rightarrow$ $ M \ddot u + C \dot u + K u = F $
\item Static problems $\rightarrow$ $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ K u = F $
\end{itemize}
%\vspace{0.4cm}
\item Existing solution methods
% \vspace*{0.5cm}
\begin{itemize}
\item \textbf{Random r.h.s} (external force random)
\begin{itemize}
\item FPK equation approach
\item Use of fragility curves with deterministic FEM (DFEM)
\end{itemize}
% \vspace*{0.2cm}
\item \textbf{Random l.h.s} (material properties random)
\begin{itemize}
\item Monte Carlo approach with DFEM $\rightarrow$ CPU expensive
% \item Stochastic finite element method (e.g. Perturbation method
% $\rightarrow$ a linearized expansion! Error increases as a function
% of COV; Spectral method
% $\rightarrow$ developed for elastic materials so far)
\item Perturbation method
$\rightarrow$ a linearized expansion! Error increases as a function
of COV
\item Spectral method
$\rightarrow$ developed for elastic materials so far
% \begin{itemize}
%
% \item Perturbation method $\rightarrow$ fails if COVs of soil $>$ 20\%
%
% \item Spectral method $\rightarrow$ only for elastic material
%
% \end{itemize}
\end{itemize}
\end{itemize}
\item Original development of {\bf Probabilistic ElastoPlasticity}
\end{itemize}
\end{frame}
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\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}
%
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\end{frame}
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% \begin{frame}
%
% \frametitle{Problem Statement}
%
%
%
%
%
% \begin{itemize}
%
% \item Incremental 3D elasticplastic stressstrain:
% %
% %
%
% \begin{equation}
% \nonumber
% \frac{ d\sigma_{ij}}{d t} = \left \{
% 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_*}
% \right \}
% \frac{ d\epsilon_{kl}}{d t}
% \end{equation}
%
%
%
%
%
% \item Focus on 1D (3D is also available) $\rightarrow$ a nonlinear ODE with random coefficient
% (material) and random forcing ($\epsilon$)
% %
% %
% %
% \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)
% \end{eqnarray}
% %
% with initial condition $\sigma(0)=\sigma_0$
%
% \end{itemize}
%
% \end{frame}
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{Solution to Probabilistic ElasticPlastic Problem}
\begin{itemize}
\item Use of stochastic continuity (Liouiville) equation (Kubo 1963)
%\vspace{0.1cm}
\item With cumulant expansion method (Kavvas and Karakas 1996)
%\vspace{0.1cm}
\item To obtain ensemble average form of Liouville Equation
%\vspace{0.1cm}
\item Which, with van Kampen's Lemma (van Kampen 1976): ensemble average of
phase density is the probability density
%\vspace{0.1cm}
\item Yields EulerianLagrangian form of the Forward Kolmogorov
(FokkerPlanckKolmogorov) equation
\end{itemize}
\end{frame}
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\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}
%
%
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\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}
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\begin{frame}
\frametitle{Probabilistic ElasticPlastic Response}
\begin{figure}[!hbpt]
\begin{center}
%\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDFedited.pdf}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2012/DOELLNLworkshop2728Feb2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_Contour_PDFedited.pdf}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{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}
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\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}
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\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}
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%\subsection{Probabilistic ElasticPlastic Response}
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\begin{frame}
\frametitle{Stochastic Finite Element Formulation}
\begin{itemize}
\item Governing equations:
\begin{equation}
\nonumber
A\sigma = \phi(t);~~~ Bu = \epsilon; ~~~\sigma = E \epsilon
\end{equation}
\vspace*{0.2cm}
\item {\bf Spatial} and
{\bf stochastic} discretization
\begin{itemize}
\vspace*{0.2cm}
\item Deterministic spatial differential operators ($A$ \& $B$) $\rightarrow$
Regular shape function method with Galerkin scheme
\vspace*{0.2cm}
\item Input random field material properties ($E$) $\rightarrow$
KarhunenLo{\`e}ve (KL) expansion, optimal expansion, error minimizing property
\vspace*{0.2cm}
\item Unknown solution random field ($u$) $\rightarrow$ Polynomial Chaos (PC)
expansion
\end{itemize}
\end{itemize}
\end{frame}
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\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}
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\begin{frame}
\frametitle{Inside SSEPFEM}
\begin{itemize}
\item Explicit stochastic elasticplastic finite element computations
\vspace*{0.2cm}
\item FPK probabilistic constitutive integration at Gauss integration points
\vspace*{0.2cm}
\item Increase in (stochastic) dimensions (KL and PC) of the problem
\vspace*{0.2cm}
\item Excellent for parallelization, both at the element and global levels
\vspace*{0.2cm}
\item Development of the probabilistic elasticplastic stiffness tensor
\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}
% 
%  \begin{frame}
% 
% 
% 
%  \frametitle{Governing Equations \& Discretization Scheme}
% 
%  \begin{itemize}
% 
%  \item Governing equations in mechanics:
% 
% 
%  \begin{equation}
%  \nonumber
%  A\sigma = \phi(t);~~~ Bu = \epsilon; ~~~\sigma = D \epsilon
%  \end{equation}
% 
%  \item Discretization (spatial and stochastic) schemes
% 
%  \begin{itemize}
% 
%  \item Input random field material properties ($D$) $\rightarrow$
%  KarhunenLo{\`e}ve (KL) expansion, optimal expansion, error minimizing property
% 
%  \item Unknown solution random field ($u$) $\rightarrow$ Polynomial Chaos (PC)
%  expansion
% 
%  \item Deterministic spatial differential operators ($A$ \& $B$) $\rightarrow$
%  Regular shape function method with Galerkin scheme
% 
% 
%  \end{itemize}
% 
% 
% 
%  \end{itemize}
% 
%  \end{frame}
% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %  %  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  % 
%  %  \begin{frame}
%  % 
%  % 
%  %  \frametitle{Truncated KarhunenLo{\`e}ve (KL) expansion}
%  % 
%  %  \begin{itemize}
%  % 
%  %  \item Representation of input random fields in eigenmodes of covariance kernel
%  % 
%  %  % \vspace*{0.1cm}
%  %  % \begin{figure}[!hbpt]
%  %  \begin{flushleft}
%  %  \includegraphics[height=3.0cm]{/home/jeremic/tex/works/Conferences/2008/GeoCongress/Probabilistic/Paper/TypicalDataPlotBH1Edited.jpg}
%  %  % ShearStrengthProfile.jpg}
%  %  \end{flushleft}
%  %  \vspace*{3.15cm}
%  %  \begin{flushright}
%  %  % \begin{equation}
%  %  % \nonumber
%  %  % \begin{normalsize}
%  %  $ q_T(x,\theta) = \bar q_T(x) + \sum_{n=1}^M \sqrt{\lambda_n} \xi_n(\theta) f_n(x) $ \\
%  %  \ \\
%  %  $ \int_D C(x_1, x_2) f (x_2) dx_2 = \lambda f (x_1) \ \ \ \ \ \ \ \ \ \ \ \ $ \\
%  %  \ \\
%  %  $ \xi_i(\theta) = \displaystyle \frac{1}{\sqrt \lambda_i} \int_D \left [q_T(x,\theta)  \bar q_T (x) \right] f_i (x) dx $
%  %  % \end{equation}
%  %  % \end{normalsize}
%  %  \end{flushright}
%  %  % \end{figure}
%  % 
%  %  % \vspace{6.0cm}
%  %  % \begin{flushright}
%  %  % \begin{equation}
%  %  % \nonumber
%  %  % w(x,\theta) = \bar w(x) + \sum_{n=0}^M \sqrt{\lambda_n} \zeta_n(\theta) f_n(x)
%  %  % \end{equation}
%  %  % \end{flushright}
%  % 
%  %  % \vspace*{0.8cm}
%  %  \item Error minimizing property
%  % 
%  %  \item Optimal expansion $\rightarrow$ minimization of number of stochastic dimensions
%  % 
%  % 
%  %  \end{itemize}
%  % 
%  % 
%  % 
%  %  \end{frame}
%  % 
%  % 
%  %  %  %  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %  %  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %  %  \begin{frame}
%  %  %  \frametitle{KL Expansion (of Covariance Kernel)}
%  %  % 
%  %  %  \begin{flushleft}
%  %  %  %\begin{center}
%  %  %  %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%  %  %  \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/ActualExponentialCovarianveSurface.jpg}
%  %  %  \hspace*{0.3cm}
%  %  %  \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_1Term_CovarianveSurface.jpg}
%  %  %  %\vspace*{1.0cm}
%  %  %  %\mbox{Exact covariance surface}
%  %  %  %\vspace*{4.0cm}
%  %  %  \end{flushleft}
%  %  %  \vspace*{0.4cm}
%  %  %  \small{\ \ \ \ \ \ \ \ \ \ \ \ \ Exact \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 term approx.} \\
%  %  %  \small{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (8.49\% error)}
%  %  %  %\begin{flushright}
%  %  %  %\end{center}
%  %  %  %\end{figure}
%  %  % 
%  %  %  %\vspace*{0.5cm}
%  %  %  %\small{Exact covariance surface \ \ \ \ \ \ \ \ \ \ \ \ \ \ Oneterm approximation}
%  %  %  %
%  %  %  %\vspace*{0.5cm}
%  %  %  %
%  %  %  %\begin{figure}[!hbpt]
%  %  %  %\begin{center}
%  %  %  \begin{flushleft}
%  %  %  \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_2Terms_CovarianveSurface.jpg}
%  %  %  \hspace*{0.2cm}
%  %  %  \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_3Terms_CovarianveSurface.jpg}
%  %  %  \end{flushleft}
%  %  %  %\vspace*{0.5cm}
%  %  %  \small{\ \ \ \ \ 2 terms approx. \ \ \ \ \ \ \ \ \ 3 terms approx.} \\
%  %  %  \small{\ \ \ \ \ \ (1.15\% error) \ \ \ \ \ \ \ \ \ \ \ \ \ (1.13\% error)}
%  %  %  %
%  %  %  \vspace{6.5cm}
%  %  %  \begin{flushright}
%  %  % 
%  %  %  \includegraphics[height=2.0cm]{/home/jeremic/tex/works/Reports/2006/SEPFEM/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FiniteScaleEdited.jpg} \hspace*{0.7cm}
%  %  %  %ShearStrengthProfile.jpg}
%  %  % 
%  %  %  covariance function \\
%  %  %  (exponential): \ \ \ \ \ \\
%  %  %  %\ \\
%  %  %  $ C(x_1, x_2) = \sigma^2 e^{ x_1  x_2  /b} $ \hspace*{0.5cm} \\
%  %  %  \ \\
%  %  %  \ \\
%  %  %  KL approximation: \ \ \ \ \ \ \ \ \\
%  %  %  \ \\
%  %  %  $ C(x_1, x_2) \ \ \ \ \ \ \ \ \ \ \ \ \ $ \\
%  %  %  $ = \sum_{k =1}^M \lambda_k f_k(x_1) f_k(x_2) $
%  %  % 
%  %  %  \end{flushright}
%  %  %  %\end{figure}
%  %  % 
%  %  %  %\vspace*{2.5cm}
%  %  %  %\small{Twoterms approximation \ \ \ \ \ \ \ \ \ \ \ \ Threeterms approximation}
%  %  % 
%  %  %  \vspace*{4.0cm}
%  %  %  \begin{center}
%  %  %  \large{KL Expansion of Covariance Kernel}
%  %  %  \end{center}
%  %  % 
%  %  %  \end{frame}
%  %  % 
%  %  % 
%  %  %  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  % 
%  %  \begin{frame}
%  % 
%  %  \frametitle{Polynomial Chaos (PC) Expansion}
%  % 
%  %  %\begin{itemize}
%  % 
%  %  %\vspace*{0.35cm}
%  %  %\item Solution (displacement) random field $\rightarrow$ Can not use KL expansion directly
%  % 
%  %  \begin{itemize}
%  % 
%  %  \item Covariance kernel is not known a priori
%  % 
%  %  \vspace*{0.5cm}
%  %  \begin{normalsize}
%  %  \begin{equation}
%  %  \nonumber
%  %  u(x,\theta)=\sum_{j=1}^L e_j \chi_j(\theta)b_j(x)
%  %  \end{equation}
%  %  \end{normalsize}
%  % 
%  %  \vspace*{0.3cm}
%  %  \item Can be expressed as functional of known random variables and unknown deterministic function
%  % 
%  %  \vspace*{0.5cm}
%  %  \begin{normalsize}
%  %  \begin{equation}
%  %  \nonumber
%  %  u(x,\theta)=\zeta[\xi_i(\theta),x]
%  %  \end{equation}
%  %  \end{normalsize}
%  % 
%  %  \vspace*{0.5cm}
%  %  \item Need a basis of known random variables $\rightarrow$ PC expansion
%  % 
%  %  \vspace*{0.2cm}
%  %  \begin{normalsize}
%  %  \begin{equation}
%  %  \nonumber
%  %  \chi_j(\theta)=\sum_{i=0}^P\gamma_i^{(j)}\psi_i\left[\left\{\xi_r\right\}\right]
%  %  \end{equation}
%  % 
%  %  \vspace*{0.5cm}
%  %  \begin{equation}
%  %  \nonumber
%  %  u(x,\theta)=\sum_{j=1}^L \sum_{i=0}^P \gamma_i^{(j)} \psi_i[\{\xi_r\}]e_j b_j(x) = \sum_{i=0}^P \psi_i[\{\xi_r\}] d_i(x)
%  %  \end{equation}
%  %  \end{normalsize}
%  % 
%  %  % \vspace*{0.6cm}
%  %  % \item Deterministic coefficients can be found by minimizing norm of error of finite
%  %  % representation (e.g. using Galerkin scheme)
%  % 
%  % 
%  % 
%  % 
%  %  % \end{itemize}
%  %  \end{itemize}
%  % 
%  % 
%  %  %\end{itemize}
%  % 
%  %  \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} 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[\{\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} = \int_D B_n \textcolor{mycolor}{D} B_m dV \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ K'_{mnk} = \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}
% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
%  \begin{frame}
%  \frametitle{Inside SSEPFEM}
% 
%  \begin{itemize}
% 
%  \item Explicit stochastic elasticplastic finite element computations
% 
%  \vspace*{0.2cm}
%  \item FPK probabilistic constitutive integration at Gauss integration points
% 
%  \vspace*{0.2cm}
%  \item Increase in (stochastic) dimensions (KL and PC) of the problem
% 
% 
%  \vspace*{0.2cm}
%  \item Development of the probabilistic elasticplastic stiffness tensor
% 
% 
%  \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}
% 
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\subsection{Seismic Wave Propagation Through Uncertain Soils}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
%
% \frametitle{Applications}
%
%
%
%
% \begin{itemize}
%
% \vspace*{0.3cm}
% \item Stochastic elasticplastic simulations of soils and structures
%
% \vspace*{0.3cm}
% \item Probabilistic inverse problems
%
% \vspace*{0.3cm}
% \item Geotechnical site characterization design
%
% \vspace*{0.3cm}
% \item Optimal material design
%
%
% \end{itemize}
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  +
%  + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  +
%  + \begin{frame}
%  +
%  +
%  + \frametitle{Random Field Modeling of Uncertain Soil Properties}
%  +
%  + \begin{itemize}
%  +
%  + \item Finite scale model
%  +
%  + \begin{itemize}
%  +
%  + \item Short memory, finite correlation length
%  +
%  + \item Common autocovariance model $\rightarrow$ exponential, spherical, triangular, linearexponential
%  +
%  + \end{itemize}
%  +
%  + \item Fractal model
%  +
%  + \begin{itemize}
%  +
%  + \item long memory, infinite correlation length $\rightarrow$ more realistic for modeling horizontal
%  + spatial uncertainty
%  +
%  + \item 1/ftype noise process with power spectral density, $P(\omega)~=~P_0~\omega^{\gamma}$, with
%  + upper and/or lower frequency cutoff.
%  +
%  + \end{itemize}
%  +
%  + \end{itemize}
%  +
%  + \end{frame}
%  +
%  + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{Seismic Wave Propagation through Stochastic Soil}
\begin{itemize}
%\item maximizing the loglikelihood of observing the spatial data under assumed joined distribution (for finite
%scale model) or maximizing the loglikelihood of observing the periodogram estimates (for fractal model)
\item Maximum likelihood estimates
\vspace*{0.3truecm}
%\begin{figure}
\begin{flushleft}
\hspace*{1.7cm}
\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/SamplingPlanEdited.jpg}
\hspace*{0.0cm}
\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalDataPlotBH1Edited.jpg} \\
\small{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Typical CPT $q_T$}
\end{flushleft}
%\end{figure}
\vspace*{4.9truecm}
%\begin{figure}
\begin{flushright}
\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FiniteScaleEdited.jpg} \\
\vspace*{0.01truecm}
\small{Finite Scale}
\end{flushright}
%\end{figure}
\vspace*{0.02truecm}
%\begin{figure}
\begin{flushright}
\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FractalEdited.jpg} \\
\small{Fractal}
\end{flushright}
%\end{figure}
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{"Uniform" CPT Site Data}
\vspace*{0.7cm}
%\begin{figure}
\begin{center}
\includegraphics[height=6.7cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/EastWestProfileEdited.pdf}
\end{center}
%\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Random Field Parameters from Site Data}
%\begin{flushleft}
%\includegraphics[height=5.0cm]{PEER2007_3.jpg}
%\end{flushleft}
%\vspace*{0.5truecm}
\begin{itemize}
\item Soil as 12.5 m deep 1D soil column (von Mises Material)
\begin{itemize}
\item Properties (including testing uncertainty) obtained through random field modeling of CPT $q_T$
%
$\left = 4.99 ~MPa;~~Var[q_T] = 25.67 ~MPa^2; $\\
Cor. ~Length $[q_T] = 0.61 ~m; $ Testing~Error $= 2.78 ~MPa^2$
\end{itemize}
\vspace*{0.2cm}
\item $q_T$ was transformed to obtain $G$: ~~$G/(1\nu)~=~2.9q_T$
\begin{itemize}
\item Assumed transformation uncertainty = 5\%
%
$\left = 11.57MPa; Var[G] = 142.32 MPa^2$ \\
Cor.~Length $[G] = 0.61 m$
\end{itemize}
%\begin{center}
%\hspace*{1.7cm}
%\includegraphics[height=3.5cm]{Chapter9_Schematic.jpg}
%\hspace*{0.0cm}
%\includegraphics[height=3.5cm]{Chapter9_BaseDisplacement.jpg} \\
%\small{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Base Displacement}
%\end{center}
\vspace*{0.2cm}
\item Input motions: modified 1938 Imperial Valley
% \vspace*{0.2cm}
% \begin{center}
% \includegraphics[height=2.0cm]{Chapter9_BaseDisplacement.jpg}
% \end{center}
\end{itemize}
\end{frame}
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%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
%
%
% \frametitle{Seismic Wave Propagation through Stochastic Soil}
%
%
%
% \begin{figure}
% \begin{center}
% \hspace*{0.75cm}
% \includegraphics[width=9.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter9Plots/Chapter9_ElasticPlasticResponseNew.pdf}
% %\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
% \end{center}
% \end{figure}
%
% Mean$\pm$ Standard Deviation
%
%
%
% %\begin{flushleft}
% %\includegraphics[height=5.0cm]{PEER2007_3.jpg}
% %\end{flushleft}
%
% % \hspace*{1.0cm} \noindent Statistics of Top Node Displacement:
% %
% % \vspace*{0.5truecm}
% %
% % \begin{figure}
% % \begin{flushleft}
% % \hspace*{1.0cm}
% % \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_MeanNew.jpg}
% % \hspace*{0.1cm}
% % \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_SDNew.jpg}
% % \end{flushleft}
% % \end{figure}
% % \vspace*{0.5truecm}
% % \hspace*{1.0cm} \tiny{~~~~~~~~~~~~~~~~~~~~~~~~~~~~Mean~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Standard Deviation}
% %
% % \vspace*{0.3truecm}
% %
% % \begin{figure}
% % \begin{flushleft}
% % \hspace*{0.75cm}
% % \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponseNew.jpg}
% % \hspace*{0.4cm}
% % \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_COVNew.jpg}
% % \end{flushleft}
% % \end{figure}
% % \vspace*{0.3truecm}
% % \hspace*{0.5cm} \tiny{~~~~~~~Mean$\pm$ Standard Deviation~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~COV}
% %
% %
% % \vspace*{6.0cm}
% % \begin{flushright}
% % \includegraphics[height=4.5cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_PDFNewEdited.jpg} \hspace*{1.0cm}
% % \end{flushright}
% %
% \end{frame}
%
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\begin{frame}
\frametitle{Decision About Site (Material) Characterization}
\begin{itemize}
\item Do nothing about site characterization (rely on experience): conservative
{\bf guess} of soil data, $COV = 225$\%, correlation length $= 12$m.
\vspace*{0.3cm}
\item Do better than standard site characterization: $COV = 103$\%, correlation
length $= 0.61$m)
\vspace*{0.3cm}
\item Improve site (material) characterization if probabilities of exceedance are unacceptable!
\end{itemize}
\end{frame}
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\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}
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\begin{frame}
\frametitle{Evolution of Mean $\pm$ SD for Guess Case}
\begin{figure}
\begin{center}
\hspace*{0.75cm}
\includegraphics[width=10.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/Evolutionary_Mean_pm_SD_NoDataEdited.pdf}
\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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 Exceedance of $20$cm}
%
%
%
% \begin{figure}
% \begin{center}
% %\hspace*{0.75cm}
% \includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/ProbabilityOfExceedance20cm_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 Exceedance of $50$cm}
%
%
%
% \begin{figure}
% \begin{center}
% %\hspace*{0.75cm}
% \includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/ProbabilityOfExceedance50cm_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}
%
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%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Summary}
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\frametitle{Summary}
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% \item High fidelity
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\item {\bf Interplay} of {\bf Uncertain} {\bf
Earthquake}, {\bf Uncertain} {\bf Soil/Rock},
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and {\bf Uncertain} {\bf Structure} in time domain {\bf probably} plays a
decisive role in seismic performance of structures (NPPs, etc.)
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\item Improve {\bf risk informed decision making} through high
fidelity {\bf Deterministic} and {\bf Stochastic ElasticPlastic Finite
Element} modeling and simulation
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\item {\bf Education and training} of users will prove essential
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\item {\bf Acknowledgement:} funding and collaboration with the USNRC, and
funding from NSF, DOE, CNSC
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\end{document}