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%\title{NRC Staff Capacity Building: \\
%       Micromechanical Origins of Elasto-Plasticity }
\title{Aspects of Deterministic and Probabilistic   \\
       Modeling and Simulation in Earthquake Engineering  }


%\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.
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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 }
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{\small LBNL Seminar, December 2012}

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  \titlepage
\end{frame}

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\begin{frame}
  \frametitle{Outline}
  \tableofcontents
  % You might wish to add the option [pausesections]
\end{frame}

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% Structuring a talk is a difficult task and the following structure
% may not be suitable. Here are some rules that apply for this
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% - At *most* three subsections per section.
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%   15 and 30 frames, all told.

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%   enough. Leave out details, even if it means being less precise than
%   you think necessary.
% - If you omit details that are vital to the proof/implementation,
%   just say so once. Everybody will be happy with that.


  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Motivation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Solution}

\begin{itemize}
  


%\vspace*{0.3cm}
  \item    {\bf Physics based  modeling  and  simulation} of seismic behavior of
  soil-structure 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 soil-structure NPP system

\vspace*{0.1cm}
  \item  {\bf  Directing},      in   space   and   time,  with {\bf high (known)
  confidence}, seismic energy flow in the soil-foundation-structure system

%\vspace*{0.1cm}
%  \item  {\bf Education} for researchers, professional practice.
  

\end{itemize}
\end{frame}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{NRC ESSI Simulator System}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - \begin{frame} 
% - \frametitle{Project Goals}
% - 
% - \begin{itemize}
% - 
% - %\vspace*{0.5cm}
% - \item    Development    of    the    NRC ESSI Simulator System 
% - for Hi-Fi modeling and simulation of non-linear earthquake 
% - soil/rock structure interaction problems:
% -   \begin{itemize}
% - %    \item Time domain, nonlinear, parallel finite element program:
% -     \item  {\bf NRC-ESSI-Program}
% - %    \item  High  performance, parallel computer: 
% -     \item  {\bf NRC-ESSI-Computer}
% - %    \item  Educational endeavor, documentation:  
% -     \item  {\bf NRC-ESSI-Notes}
% -   \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}
% - 
% - 
% - 
% - 
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{NRC ESSI Simulator System}

\begin{itemize}

\item {\bf The NRC-ESSI-Program} is a 3D, nonlinear, time domain, 
parallel finite element program specifically developed for
Hi-Fi modeling  and  simulation  of  Earthquake  Soil/Rock  Structure
Interaction problems for NPPs on NRC-ESSI-Computer. \
%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 (NRC-ESSI-Program). Significant effort is devoted to development
%of  verification  and  validation  procedures,  as  well  as  on  development of
%extensive  documentation.  NRC-ESSI-Program  is  in  public  domain  and is licensed
%through the Lesser GPL.

%\vspace*{0.3cm}
\vspace*{0.1cm}
\item {\bf The NRC-ESSI-Computer} 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    NRC-ESSI-Notes} 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
%NRC-ESSI-Program  code  API (application Programming Interface) and DSLs (Domain
%Specific Language).
%%NRC-ESSI-Notes,    developed  by Boris Jeremic and collaborators, are in public
%domain
%%and  are  licensed  under a Creative Commons Attribution-ShareAlike 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://nrc-essi-simulator.info}}


\end{itemize}

\end{frame}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Collection of Useful Libraries (Modeling Part)}

\begin{itemize}


\vspace*{0.2cm}
\item  Template3D-EP  libraries   for  elastic  and elastic-plastic
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, un-correlated 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, GPL-like) 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 cost-performance \\
         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: \verb|2.6.35.10-74.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: ConnectX-2 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 1810-48G 48 Port


  \item  InfiniBand::  Mellanox  MIS5030Q-1SFCA 36-port 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,  EL-PL, 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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%











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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}-Foundation-Structure Systems}
 \frametitle{High Fidelity Modeling}
% \frametitle{Seismic Energy Dissipation for 
%             \underline{Soil}-Foundation-Structure 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}
% - 
% - 
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - 
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - \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}
% - 
% - \end{frame}
% - 
% - 
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - 
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\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 Elastic-plastic 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}






















%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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, 8-20, 27 node element)
  \item  Two phase (fully coupled, porous solid, pore fluid) solid bricks (8 and
  27 node:   $u-p-U$, $u-p$)

\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}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Material Models for Solids and Structures}

    \begin{itemize}
      \item Elastic: linear, nonlinear isotropic, cross anisotropic
      
\vspace*{0.3cm}
      \item        Elastic-Plastic:    von   Mises,  Drucker--Prager, 
      Cam--Clay, Rounded Mohr--Coulomb, Parabolic Leon, 
      SANIsand (Dafalias--Manzari...), 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, elastic-plastic)
% 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_e-K^{\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/NRC-LBL-ProjectReviewMeeting_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 elastic-plastic

  \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/NRC-LBL-ProjectReviewMeeting_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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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*{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, elastic-plastic...)


   \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: 10-12 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, elastic-plastic ?)
%  
%    \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}{|c|c|c|c|c|}
\begin{tabular}{|r|m{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}c|m{2.8cm}c|m{2.8cm}c|m{3.0cm}c|m{4.0cm}c|}
% \begin{tabularx}{\linewidth}{|c|c|c|c|c|}
% \begin{tabular*}{0.75\textwidth}{@{\extracolsep{\fill}}|c|c|c|c|c|}
\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 
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% Ex out \frametitle{Acc. 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.pdf}
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% 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}
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% Ex out %\caption{Comparison of acceleration time histories of the structure between 
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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}
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% 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}
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% Ex out %\caption{Comparison of FFT of the acceleration of the structure between 
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% Ex out \frametitle{Gaping Response ($45^\circ$ Ricker Wavelet)}
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% Ex out $4.5s$
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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 %\caption{Distribution of gap openings along the contact interface for Ricker wave
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% Ex out \frametitle{Slipping Response and Energy Dissipated ($45^\circ$ Ricker)}
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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}
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% Ex out %\caption{Distribution of sliding along the contact interface for Ricker wave
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% Ex out   \frametitle{Passive Base Isolation in Uniform and Layered Soils}
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% Ex out %\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
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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 Point-wise \\ 
              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 point-wise})
% 
%       \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}
% - 
% - 
% - \end{frame}
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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 trade-off 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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\end{frame}







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\begin{frame}


\frametitle{Recent State-of-the-Art}

\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 Elasto--Plasticity}


\end{itemize}

\end{frame}



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\begin{frame} 

\frametitle{Uncertainty Propagation through Constitutive Eq.}

% 

\begin{itemize}

 


\item Incremental el--pl 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 elastic--plastic}}
%
           \end{array} \right.
           \end{eqnarray}

%\end{normalsize}

 



%\vspace{0.5cm}
% \item Non--linear coupling in the El--Pl 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 1-D $\rightarrow$ 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 elastic-plastic stress--strain:
% % 
% %    
% 
%     \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}
% 
% 
% 
% 
 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 


\frametitle{Solution to Probabilistic Elastic-Plastic 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      Eulerian-Lagrangian  form  of the Forward Kolmogorov
(Fokker-Planck-Kolmogorov) equation

\end{itemize}
 
\end{frame}







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\begin{frame} 

\frametitle{Probabilistic Stress Solution: \\ Eulerian--Lagrangian 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 3-D probabilistic stress-strain response
%-- 
%-- \end{itemize}
%-- 
%--  


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 

\frametitle{Eulerian--Lagrangian FPK Equation and (SEP)FEM}

 

\begin{itemize}

\item Advection-diffusion 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 Second-order  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 elastic-plastic-damage 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 Elastic-Plastic Response}


\begin{figure}[!hbpt]
\begin{center}
%\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDF-edited.pdf}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2012/DOE-LLNL-workshop-27-28-Feb-2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_Contour_PDF-edited.pdf}
\end{center}
\end{figure}

\end{frame}

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\begin{frame} 

\frametitle{Probabilistic Elastic-Plastic Response}


\begin{figure}[!hbpt]
\begin{center}
%\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Conferences/2011/ICASP11_Zurich/Present/PDF_Plot-Ed.pdf}
\includegraphics[width=9.5cm]{/home/jeremic/tex/works/Conferences/2012/DOE-LLNL-workshop-27-28-Feb-2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_PDF-edited.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/JGGE-GoverGmax/figures/ShearStrength_RawData_and_MeanTrend-Mod.pdf}
\hfill
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/ShearStrength_Histogram_PearsonIV-FineTuned-Mod.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/JGGE-GoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01-Ed.pdf} 
\hfill
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/YoungModulus_Histogram_Normal_01-Ed.pdf}
%
\end{center}
\end{figure}                                                          

\vspace*{-0.3cm}
 Transformation of SPT $N$-value $\rightarrow$ 1-D 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{Probabilistic Elastic--Plastic 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   ($E$)   $\rightarrow$
Karhunen--Lo{\`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}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Spectral Stochastic Elastic--Plastic 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 Fokker--Planck--Kolmogorov 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 KL-expansion
%      
%      \item Representation of (unknown) solution random variables using polynomial chaos of
%            (known) input random variables
%
%      \item Fokker--Planck--Kolmogorov 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 elastic--plastic 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 elastic--plastic stiffness tensor


\end{itemize}

%  \noindent Salient Features:
       
%      \begin{itemize}
%
%      \item Efficient representation of input random fields into finite number of random
%            variables using KL-expansion
%      
%      \item Representation of (unknown) solution random variables using polynomial chaos of
%            (known) input random variables
%
%      \item Fokker--Planck--Kolmogorov 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   ($D$)   $\rightarrow$
% -  Karhunen--Lo{\`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 Karhunen--Lo{\`e}ve (KL) expansion}
% -  % -               
% -  % -        \begin{itemize}                                                                                                                         
% -  % -                                                                                                                                                       
% -  % -        \item Representation of input random fields in eigen-modes of covariance kernel                                                                
% -  % -                                                                                                                                                       
% -  % - %       \vspace*{-0.1cm}                                                                                                                               
% -  % - %              \begin{figure}[!hbpt]                                                                                                                   
% -  % -               \begin{flushleft}                                                                                                                       
% -  % -               \includegraphics[height=3.0cm]{/home/jeremic/tex/works/Conferences/2008/GeoCongress/Probabilistic/Paper/TypicalDataPlotBH1-Edited.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 \ \ \ \ \ \ \ \ \ \ \ \ \ \ One-term 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_FiniteScale-Edited.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{Two-terms approximation \ \ \ \ \ \ \ \ \ \ \ \  Three-terms 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 K-L 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 Elastic--Plastic 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 Fokker--Planck--Kolmogorov 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 KL-expansion
% -  %      
% -  %      \item Representation of (unknown) solution random variables using polynomial chaos of
% -  %            (known) input random variables
% -  %
% -  %      \item Fokker--Planck--Kolmogorov 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 elastic--plastic 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 elastic--plastic stiffness tensor
% -  
% -  
% -  \end{itemize}
% -  
% -  %  \noindent Salient Features:
% -         
% -  %      \begin{itemize}
% -  %
% -  %      \item Efficient representation of input random fields into finite number of random
% -  %            variables using KL-expansion
% -  %      
% -  %      \item Representation of (unknown) solution random variables using polynomial chaos of
% -  %            (known) input random variables
% -  %
% -  %      \item Fokker--Planck--Kolmogorov 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 elastic--plastic 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}
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% - +
% - + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - + 
% - + \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, linear--exponential
% - + 
% - + \end{itemize}
% - + 
% - + \item Fractal model 
% - + 
% - + \begin{itemize}
% - + 
% - + \item long memory, infinite correlation length $\rightarrow$ more realistic for modeling horizontal 
% - + spatial uncertainty
% - + 
% - + \item 1/f-type noise process with power spectral density, $P(\omega)~=~P_0~\omega^{-\gamma}$, with
% - + upper and/or lower frequency cut-off.
% - + 
% - + \end{itemize}
% - + 
% - + \end{itemize}
% - + 
% - + \end{frame}
% - + 
% - + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}


\frametitle{Seismic Wave Propagation through Stochastic Soil}


\begin{itemize}

%\item maximizing the log--likelihood of observing the spatial data under assumed joined distribution (for finite 
%scale model) or maximizing the log--likelihood 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/SamplingPlan-Edited.jpg}
\hspace*{0.0cm}
\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalDataPlotBH1-Edited.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_FiniteScale-Edited.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_Fractal-Edited.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/EastWestProfile-Edited.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 1--D soil column (von Mises Material) 

\begin{itemize}

\item Properties (including testing uncertainty) obtained through random field modeling of CPT $q_T$
%
$\left<q_T\right> = 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\%
%
$\left<G\right> = 11.57MPa; Var[G] = 142.32 MPa^2$ \\
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_ElasticPlasticResponse-New.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}
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%--  % \hspace*{-1.0cm} \tiny{~~~~~~~~~~~~~~~~~~~~~~~~~~~~Mean~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Standard Deviation}
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%--  % \hspace*{-0.5cm} \tiny{~~~~~~~Mean$\pm$ Standard Deviation~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~COV}
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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 Elastic-Plastic\\
        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/UNION-Univ-BGD/Present/Plots_with_Labels/EvolutionaryPDF_Actual-Edited.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/UNION-Univ-BGD/Present/Plots_with_Labels/Evolutionary_Mean_pm_SD_NoData-Edited.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}

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\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/UNION-Univ-BGD/Present/Plots_with_Labels/PDFs_at6sec_Actual_vs_NoData-Edited.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}

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\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/UNION-Univ-BGD/Present/Plots_with_Labels/CDFs_at6sec_Actual_vs_NoData-Edited.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}

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\end{frame}

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% 
% 
% \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/UNION-Univ-BGD/Present/Plots_with_Labels/ProbabilityOfExceedance20cm_Actual_vs_NoData-Edited.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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% 
% \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/UNION-Univ-BGD/Present/Plots_with_Labels/ProbabilityOfExceedance50cm_Actual_vs_NoData-Edited.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{Probabilities of Exceedance vs. Displacements}
% 
% 
% 
% \begin{figure}
% \begin{center}
% %\hspace*{-0.75cm}                                                                                               
% \includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/Plots_with_Labels/ProbabilityOfExceedance_vs_Displacement_Actual_vs_NoData-Edited.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}
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% % 
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% 
% \begin{frame}
% 
% \frametitle{Probabilities of Unacceptable Deformation}
% 
% \begin{figure}
% \begin{center}
% \vspace*{-0.3cm}                                                                                               
% \includegraphics[width=10.5cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/NewPlots/with_legends_and_labels/Exceedance20cm_LomaPrieta-Edited_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}
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
                                        
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\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/UNION-Univ-BGD/Present/NewPlots/with_legends_and_labels/Exceedance50cm_LomaPrieta-Edited_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}

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\begin{frame}

\frametitle{Risk Informed Decision Process}

\begin{figure}
\begin{center}
%\hspace*{-0.75cm}                                                                                               
\includegraphics[width=8.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/NewPlots/with_legends_and_labels/Summary_LomaPrieta-Edited.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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\section{Summary}



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\begin{frame}
  \frametitle{Summary}
  
  \begin{itemize}


%  \item  High fidelity
%  modeling and simulations for performance assessment of infrastructure systems


\vspace*{0.2cm}
  \item              {\bf    Interplay}        of     {\bf Uncertain} {\bf 
  Earthquake}, {\bf Uncertain}  {\bf Soil/Rock},
%  {Foundation} 
  and  {\bf  Uncertain}   {\bf Structure} in  time domain {\bf probably} plays a
  decisive role in seismic performance of structures (NPPs, etc.)

\vspace*{0.2cm}
  \item           Improve {\bf risk informed decision making}  through  high 
  fidelity  {\bf  Deterministic}  and  {\bf  Stochastic  Elastic-Plastic  Finite
  Element} modeling and simulation

 
\vspace*{0.2cm}
  \item  {\bf Education and training} of users will prove essential
  

\vspace*{0.2cm}
  \item    {\bf Acknowledgement:} funding and collaboration with the US-NRC, and
  funding from NSF, DOE, CNSC


  
\end{itemize}


\end{frame}
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\end{document}


