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\title{Verification Procedures for \\
       Simulation of Fully Coupled Behavior of \\
       Porous Media}


%\subtitle
%{Include Only If Paper Has a Subtitle}

%\author[Author, Another] % (optional, use only with lots of authors)
%{F.~Author\inst{1} \and S.~Another\inst{2}}
% - Give the names in the same order as the appear in the paper.
% - Use the \inst{?} command only if the authors have different
%   affiliation.
\pgfdeclareimage[height=0.2cm]{university-logo}{/home/jeremic/BG/amblemi/ucdavis_logo_blue_sm}
\pgfdeclareimage[height=0.7cm]{lbnl-logo}{/home/jeremic/BG/amblemi/lbnl-logo}

\author[Jeremi{\'c}] % (optional, use only with lots of authors)
{Boris~Jeremi{\'c},  Panagiota Tasiopoulou, Mahdi Taiebat, Nima Tafazzoli, Mario
Martinelli}
%{Boris~Jeremi{\'c},  Nima  Tafazzoli,  Babak Kamrani, Panagiota Tasiopoulou and
%Chang-Gyun Jeong}
% - Give the names in the same order as the appear in the paper.
% - Use the \inst{?} command only if the authors have different
%   affiliation.


%\institute[Computational Geomechanics Group \hspace*{0.3truecm}
\institute[\pgfuseimage{university-logo}\hspace*{0.1truecm}\pgfuseimage{lbnl-logo}] % (optional, but mostly needed)
{UCD, LBNL, NTUA, UBC, UCD, URoma}
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.


\date[] % (optional, should be abbreviation of conference name)
{\small USNCCM11, Minneapolis, MN}

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\end{frame}

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


% Structuring a talk is a difficult task and the following structure
% may not be suitable. Here are some rules that apply for this
% solution:

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

% - A conference audience is likely to know very little of what you
%   are going to talk about. So *simplify*!
% - In a 20min talk, getting the main ideas across is hard
%   enough. Leave out details, even if it means being less precise than
%   you think necessary.
% - If you omit details that are vital to the proof/implementation,
%   just say so once. Everybody will be happy with that.

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\section{Introduction}
                         
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\begin{frame}
 \frametitle{Introduction}

\begin{itemize}

  \item Numerical analysts, designers need the best available tools
  for performance assessment (numerical predictions)

  \item Verification and validation process ensures accuracy of numerical predictions 

  \item How much can (should) we trust  model implementations (verification)?

  \item How much can (should) we trust numerical simulations (validation)?

  \item How good are our numerical predictions?

  \item The T experiments


  \item Focus on verification


\end{itemize}
\end{frame}




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\section{Verification and Validation}
%\subsection{Saturated Soils}



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\subsection{Verification}




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\begin{frame}
 \frametitle{Verification, Validation, Prediction}


\begin{itemize}

     \item 
%     Verification: 
% the  process  of  determining  that    a  model
%   implementation  accurately  represents the developer's conceptual description
%   and specification. Mathematics issue. 
{ 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. 
{ 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}

\end{frame}
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\begin{frame}
 \frametitle{Role of Verification and Validation}


\begin{figure}[!h]
\begin{center}
\hspace*{-2cm}
{\includegraphics[width=6.0cm]{/home/jeremic/tex/works/Presentation/2003/Verif_and_Valid/RoleVV.pdf}}
{\includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2011/USNCCM11_Minneapolis/Coupled/Present/VandV_ODEN.jpg}}
\hspace*{-2cm}
\end{center}
\end{figure}

{Oberkampf et al. \hspace*{4cm}  Oden et al.}
%
%\item Models available (some now, some later)
%\vspace*{-2.0cm}
\end{frame}
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\begin{frame}
  \frametitle{Verification}



   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.}



\vspace*{-0.2cm}
\begin{figure}[!h]
\begin{center}
{\includegraphics[width=10.0cm]{/home/jeremic/tex/works/Conferences/2005/OpenSeesWorkshopAugust/DeveloperSymposium/VerifValidFund01.pdf}}
\end{center}
\end{figure}


% % 
% \vspace*{-2.9cm}
% %\hspace*{0.2cm}
% \begin{figure}[!htbp] 
% \begin{flushright}
% {\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Conferences/2005/OpenSeesWorkshopAugust/DeveloperSymposium/VerifValidFund01.pdf}}
% %bridge.}
% \end{flushright}
% \end{figure}
% \hspace*{-0.5cm}
% \vspace*{-2.0cm}



\end{frame}

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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%   \frametitle{Verification}
% 
%    Mathematics issue. {\it Verification provides evidence that the
%   model is solved correctly.}
% 
% \vspace*{0.3cm}
%    \begin{itemize}
%      \item Identify and remove errors in computer coding
%        \begin{itemize}
%          \item Numerical algorithm verification
%          \item Software quality  assurance practice
%        \end{itemize}
% 
% \vspace*{0.3cm}
%      \item Quantification of the  numerical errors in  computed solution
%    \end{itemize}
% 
% 
% 
% \end{frame}
% 
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\section{Saturated Soils}
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\subsection{Fully Coupled Formulation}

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\begin{frame}
 \frametitle{Dynamic Equilibrium for Saturated, Coupled Systems}

\begin{itemize}
  \item Effective stress principle 
  $\sigma^{\prime}_{ij}  =  \sigma_{ij}  +  \alpha \delta_{ij} p$ ; ($p=-1/3 \sigma_{kk}$)

\vspace*{0.3cm}
  \item Equilibrium of the mixture 
  $ \sigma_{ij,j}-\rho \ddot{u}_i-\rho_f[\ddot{w}_i+
  \underline{\dot{w}_j\dot{w}_{i,j}}]+\rho          b_i=0$          ;         ($
  \rho=n\rho_f+(1-n)\rho_s$)
\vspace*{0.3cm}
  \item Equilibrium of the fluid
  $-p_{,i}- R_i - \rho_f \ddot{u}_i-\rho_f[\ddot{w}_i+
\underline{\dot{w}_j \dot{w}_{i,j}}]/n+\rho_f b_i=0$;
(Darcy: 
$n\dot{w}_j = Ki$; $i=h_{,j}$;  
$R_i=k_{ij}^{-1} \dot{w}_j$;  
$k_{ij}=K_{ij}/\rho_f g$ $[m]^3[s]/[kg]$)

\vspace*{0.3cm}
  \item Flow conservation 
  $\dot{w}_{i,i}+\alpha       \dot{\varepsilon}_{ii}+{\dot{p}}/{Q}+\underline{n
{\dot{\rho_f}}/{\rho_f}+\dot{s}_0}=0$; 
${1}/{Q}\equiv {n}/{K_f}+({1-n})/{K_s}$

\end{itemize}
%\vspace*{-2.0cm}
\end{frame}




% 
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%  \begin{frame}
%   \frametitle{Dynamic Equilibrium for Saturated, Coupled Systems (cont.)}
%  
%  After  neglecting  convective  accelerations,  density  variations  and assuming
%  isothermal process (no volume expansion):
%  
%  \begin{itemize}
%    \item Equilibrium of the mixture \\
%    $ \sigma_{ij,j}-\rho \ddot{u}_i-\rho_f \ddot{w}_i+\rho b_i=0$  
%  
%  \vspace*{0.3cm}
%    \item Equilibrium of the fluid \\
%    $-p_{,i}- R_i - \rho_f \ddot{u}_i-\rho_f\ddot{w}_i/n+\rho_f b_i=0$
%  
%  \vspace*{0.3cm}
%    \item Flow conservation \\
%    $\dot{w}_{i,i}+\alpha  \dot{\varepsilon}_{ii}+{\dot{p}}/{Q}=0$
%  
%  \end{itemize}
%  %\vspace*{-2.0cm}
%  \end{frame}
%  
%  
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame}
%   \frametitle{Dynamic Equilibrium for Coupled Systems (cont.)}
%  
%  Replace   relative      pseudo--displacement   $w_i$   with   real  displacement
%  $U_i=u_i+U_i^R=u_i+{w_i}/{n}$
%  
%   
%  \begin{figure}[!hbpt]
%  \begin{center}
%  \includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/LectureNotes/Figures/Darcy_vs_Real_01.pdf}
%  \end{center}
%  \end{figure}
%  
%  
%  %\vspace*{-2.0cm}
%  \end{frame}
%  
%  
%  
%  
%  
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame}
%   \frametitle{Dynamic Equilibrium for Coupled Systems (cont.)}
%  
%  
%  After  some manipulations we obtain
%  
%  
%  \begin{eqnarray*}
%  \sigma_{ij,j}^{''}-(\alpha-n) p_{,i}+(1-n) \rho_s b_i-(1-n) \rho_s \ddot{u}_i +
%  n R_i=0
%  \label{34}
%  \end{eqnarray*}
%  
%  \begin{eqnarray*}
%  -n p_{,i}+n \rho_f b_i-n \rho_f \ddot{U}_i
%  - n R_i=0
%  \label{35}
%  \end{eqnarray*}
%  
%  \begin{eqnarray*}
%  -n \dot{U}_{i,i}=(\alpha-n) \dot{\varepsilon}_{ii}+ \dot{p}/Q
%  \label{36}
%  \end{eqnarray*}
%  
%  
%  %\vspace*{-2.0cm}
%  \end{frame}
%  
%  
%  
%  
%  
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\subsection{Fully Coupled Formulation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Fully Coupled $u-p-U$ Formulation}

\begin{itemize}
  \item Formulation: fully coupled by Zienkiewicz and Shiomi 1984),
  nonlinear dynamics  by Argyris and Mlejnek (1991) 
\vspace*{0.3cm}
  \item  Physical,  velocity  proportional  damping from solid--fluid interaction
  (not using Rayleigh damping)
\vspace*{0.3cm}
  \item Accelerations of pore fluid not neglected 
    \begin{itemize}
      \item important for SFSI
      \item inertial forces of fluid allow liquefaction modeling
    \end{itemize}
\vspace*{0.3cm}
  \item Stable formulation for near incompressible pore fluid
%  \item  Formulation and implementation verified on a number of available closed
%  form solutions

\end{itemize}
%\vspace*{-2.0cm}
\end{frame}


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\begin{frame}
 \frametitle{Finite Element Discretization}

\begin{small}
\begin{eqnarray*}
& &\left[ \begin{array}{ccc}
(M_s)_{KijL} & 0 & 0 \\
 0  & 0 & 0 \\
 0  & 0 & (M_f)_{KijL}
\end{array} \right]
\left[ \begin{array}{c}
\ddot{\overline{u}}_{Lj} \\
\ddot{\overline{p}}_N \\
\ddot{\overline{U}}_{Lj}
\end{array} \right]
+
\nonumber\\
+
& &\left[ \begin{array}{ccc}
(C_1)_{KijL} & 0 & -(C_2)_{KijL} \\
 0  & 0 &   0  \\
-(C_2)_{LjiK} & 0 & (C_3)_{KijL} \\
\end{array} \right]
\left[ \begin{array}{c}
\dot{\overline{u}}_{Lj} \\
\dot{\overline{p}}_N \\
\dot{\overline{U}}_{Lj}
\end{array} \right]    
+
\nonumber\\
+
& &\left[ \begin{array}{ccc}
(K^{EP})_{KijL} & -(G_1)_{KiM} & 0 \\
-(G_1)_{LjM}  & -P_{MN} & -(G_2)_{LjM} \\
 0  & -(G_2)_{KiL} & 0
\end{array} \right]
\left[ \begin{array}{c}
\overline{u}_{Lj} \\
\overline{p}_M \\
\overline{U}_{Lj}
\end{array} \right]
=
\left[ \begin{array}{c}
\overline{f}_{Ki}^{solid} \\
0  \\
\overline{f}_{Ki}^{fluid}
\end{array} \right] \nonumber\\
\label{68}
\end{eqnarray*}
\end{small}


\end{frame}

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\begin{frame}
 \frametitle{Finite Element Discretization}

 %
\begin{small}
\begin{eqnarray*}
(M_s)_{KijL}      =\int_{\Omega}  N_K^u (1-n) \rho_s \delta_{ij} N_L^u d\Omega
\;\; &\mbox{;}& \;\;
(M_f)_{KijL}   =\int_{\Omega} N_K^U n \rho_f \delta_{ij} N_L^U d\Omega 
%%%%%%%%
\\
%%%%%%%%
(C_1)_{KijL}   =\int_{\Omega} N_K^u n^2 k_{ij}^{-1} N_L^u d\Omega
\;\; &\mbox{;}& \;\;
(C_2)_{KijL}   =\int_{\Omega} N_K^u n^2 k_{ij}^{-1} N_L^U d\Omega 
%%%%%%%%
\\
%%%%%%%%
(C_3)_{KijL}   =\int_{\Omega} N_K^U n^2 k_{ij}^{-1} N_L^U d\Omega
\;\; &\mbox{;}& \;\;
(K^{EP})_{KijL}=\int_{\Omega} N_{K,m}^u D_{imjn} N_{L,n}^u d\Omega 
%%%%%%%%
\\
%%%%%%%%
(G_1)_{KiM}    =\int_{\Omega} N_{K,i}^u (\alpha-n) N_M^p d\Omega
\;\; &\mbox{;}& \;\;
(G_2)_{KiM}    =\int_{\Omega} n N_{K,i}^U N_M^p d\Omega 
%%%%%%%%
\\
%%%%%%%%
P_{NM}         =\int_{\Omega} N_N^p \frac{1}{Q} N_M^p d\Omega
& &
\end{eqnarray*}
\end{small}
%
%
%
%
%\newpage


\end{frame}

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\begin{frame}
 \frametitle{Finite Element Discretization}

%
%
%\newpage
\begin{eqnarray*}
\overline{f}_{Ki}^{solid}
&=&
\int_{\Gamma_t} N_K^u n_j \sigma_{ij}^{''} d\Gamma
-
%
\\ & & 
%
\int_{\Gamma_p} N_K^u (\alpha-n) n_i p d\Gamma
%
 \\ & & 
%
+
\int_{\Omega} N_K^u (1-n) \rho_s b_i d\Omega 
%
\nonumber\\
~
\nonumber\\
%
\overline{f}_{Ki}^{fluid}
&=&
-
\int_{\Gamma_p} n N_K^U n_i p d\Gamma
%
\\& & 
%
+ 
\int_{\Omega} n N_K^U \rho_f b_i d\Omega 
\end{eqnarray*}
%


\end{frame}




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

     
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\section{Verification Suite}







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\subsection{Examples}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Verification Suite}


\begin{itemize}
   
   \item Code Verification
   
     \begin{itemize}


       \item Memory 
       \item Function call arguments
       \item Code  coverage
       \item Argument bounds
       \item Compiler warnings
     
     \end{itemize}

   
   
   
 \item Computational Solution Verification

   \begin{itemize}

     \item Drilling of a well [Coussy 04]
     
     \item The Case of a Spherical Cavity [Coussy 04]
     
     \item Consolidation of a Soil Layer  [Coussy 95]
     
     \item Line Injection of a fluid in a Reservoir [Coussy 95]
     
     \item Wave propagation, step displacement [Gajo and Mongiovi 95]
     
     \item Wave propagation, step velocity loading [de Boer et al. 93]
     
     \item Wave propagation, step force loading [Hiremath et al. 88]
  \end{itemize}
   


\end{itemize}

\end{frame}

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\begin{frame}
 \frametitle{Vertical Consolidation }

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\begin{figure}[!hbpt]
\begin{LARGE}
\begin{sffamily}
\begin{center} 
\includegraphics[width=4cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/consolidation.original/Figures/Self_load/all_dist.pdf}
\includegraphics[width=4cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/consolidation.original/Figures/cons/dissipation.pdf}
\\
\includegraphics[width=4cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/consolidation.original/Figures/cons/comparison_depth.pdf}
\includegraphics[width=4cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/consolidation.original/Figures/cons/settle.pdf}
\end{center}
%\caption{\label{num_model}{(a) The geometry of the soil layer of thickness h under uniform constant vertical pressure $\varpi$ applied at the surface, (b) the finite element mesh with the appropriate boundary conditions and loading at top and (c) the time history of vertical loading at top (Heaviside function).}}
\end{sffamily}
\end{LARGE}
\end{figure}

%

\end{frame}
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% - %
% - \end{frame}
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\begin{frame}
 \frametitle{Vertical Consolidation: Normalized Excess Pore Pressure}



 \vspace*{-0.5cm}
\begin{figure}[!hbpt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=10cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/consolidation.original/Figures/cons/comparison-time.pdf}
\end{center}
%\caption{\label{dissipation}{Time histories of the normalized excess pore pressure for various depths ($z=2m,5m,10m$) with respect to real time, $t$ and normalized time,$T_v$.Comparison between
%numerical and analytical results.}}
\end{sffamily}
\end{Large}
\end{figure}

%
\end{frame}






















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\begin{frame}
 \frametitle{Shock Wave Propagation, Step Displacement}


\begin{figure}[!hbpt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=0.8\textwidth]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO/GAJO_Figures/numerical_model.pdf}
\end{center}
%\caption{\label{Fig1001}{(a)   The   representative   semi-finite   soil  column
%subjected  to  a step vertical displacement equal to $1.0\times10^{-3}cm$ at the
%surface,  (b)  the  finite element mesh and the applied boundary conditions used
%for the numerical modeling and (c) the time history of the vertical displacement
%applied  at  the  top  nodes  of  the  mesh,  both to the solid and fluid phases
%(Heaviside function). }}
\end{sffamily}
\end{Large}
\end{figure}


%

\end{frame}
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Shock Wave Propagation: Step Displacement}





\vspace*{-0.5cm}
\begin{figure}[!hbpt]
\begin{center}
\hspace*{-0.8cm}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/solid_k10-8_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/solid_k10-8_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/fluid_k10-8_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/fluid_k10-8_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/solid_k10-6_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/fluid_k10-6_large.pdf}
\\
\vspace*{-0.1cm}
\hspace*{-0.8cm}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/fluid_k10-6_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/solid_k10-5_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/solid_k10-5_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/fluid_k10-5_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/fluid_k10-5_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/comparison_large_solid.pdf}
\\
\vspace*{-0.1cm}
\hspace*{-0.8cm}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/comparison_close_solid.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/comparison_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/comparison_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/solid_k_10-8_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/solid_k_10-8_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/fluid_k_10-8_large.pdf}
\\
\vspace*{-0.1cm}
\hspace*{-0.8cm}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/fluid_k_10-8_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/solid_k_10-6_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/solid_k_10-8_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/fluid_k_10-6_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/fluid_k_10-8_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/solid_k_10-5_large.pdf}
\\
\vspace*{-0.1cm}
\hspace*{-0.8cm}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/solid_k_10-5_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/fluid_k_10-5_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/fluid_k_10-5_close.pdf}
%\caption{\label{Fig1}{Time  histories of solid displacement due to longitudinal wave at the depth of 1  cm  below  the  surface.  Comparison  between numerical results (FEM) and analytical  solution  by  \cite{Gajo1995b}  for  the  case  of viscous coupling, $k$, equal to $10^{-8}  cm^3s/g$.  Two  different sets of Newmark parameters were used for the numerical analysis.}}
\end{center}
\end{figure}

\end{frame}


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%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Shock Wave Propagation: Porous Solid}





\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=10cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/comparison_large_solid.pdf}
%\caption{\label{Fig1}{Time  histories of solid displacement due to longitudinal wave at the depth of 1  cm  below  the  surface.  Comparison  between numerical results (FEM) and analytical  solution  by  \cite{Gajo1995b}  for  the  case  of viscous coupling, $k$, equal to $10^{-8}  cm^3s/g$.  Two  different sets of Newmark parameters were used for the numerical analysis.}}
\end{center}
\end{figure}

\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Shock Wave Propagation: Pore Fluid}





%\vspace*{-0.5cm}
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=10cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/comparison_large.pdf}
\end{center}
\end{figure}

\end{frame}


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\section{Summary}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Summary}

  \begin{itemize}


%\vspace*{0.3cm}
    \item Importance of verification and validation for numerical predictions 
    
\vspace*{0.3cm}
    \item Numerical predictions under uncertainty

\vspace*{0.3cm}
    \item        Would        you    trust        numerical   simulations   (for
    design/regulation/evaluation)  if  your  program of choice (simulation tool)
    did not follow (extensive) verification and validation procedures?

  \end{itemize}



\end{frame}
%--

\end{document}


