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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]{universitylogo}{/home/jeremic/BG/amblemi/ucdavis_logo_blue_sm}
\pgfdeclareimage[height=0.7cm]{lbnllogo}{/home/jeremic/BG/amblemi/lbnllogo}
\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
%ChangGyun Jeong}
%  Give the names in the same order as the appear in the paper.
%  Use the \inst{?} command only if the authors have different
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%\institute[Computational Geomechanics Group \hspace*{0.3truecm}
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{UCD, LBNL, NTUA, UBC, UCD, URoma}
%  Use the \inst command only if there are several affiliations.
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\date[] % (optional, should be abbreviation of conference name)
{\small USNCCM11, Minneapolis, MN}
\subject{}
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\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:
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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+(1n)\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
$\dot{w}_{i,i}+\alpha \dot{\varepsilon}_{ii}+{\dot{p}}/{Q}+\underline{n
{\dot{\rho_f}}/{\rho_f}+\dot{s}_0}=0$;
${1}/{Q}\equiv {n}/{K_f}+({1n})/{K_s}$
\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 pseudodisplacement $w_i$ 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}^{''}(\alphan) p_{,i}+(1n) \rho_s b_i(1n) \rho_s \ddot{u}_i +
% n R_i=0
% \label{34}
% \end{eqnarray*}
%
% \begin{eqnarray*}
% n p_{,i}+n \rho_f b_in \rho_f \ddot{U}_i
%  n R_i=0
% \label{35}
% \end{eqnarray*}
%
% \begin{eqnarray*}
% n \dot{U}_{i,i}=(\alphan) \dot{\varepsilon}_{ii}+ \dot{p}/Q
% \label{36}
% \end{eqnarray*}
%
%
% %\vspace*{2.0cm}
% \end{frame}
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\subsection{Fully Coupled Formulation}
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\begin{frame}
\frametitle{Fully Coupled $upU$ 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 solidfluid 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 (1n) \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 (\alphan) 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 (\alphan) n_i p d\Gamma
%
\\ & &
%
+
\int_{\Omega} N_K^u (1n) \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}
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\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}
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %
%  \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/comparisontime.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 semifinite soil column
%subjected to a step vertical displacement equal to $1.0\times10^{3}cm$ at the
%surface, (b) the finite element mesh and the applied boundary conditions used
%for the numerical modeling and (c) the time history of the vertical displacement
%applied at the top nodes of the mesh, both to the solid and fluid phases
%(Heaviside function). }}
\end{sffamily}
\end{Large}
\end{figure}
%
\end{frame}
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\begin{frame}
\frametitle{Shock Wave Propagation: Step Displacement}
\vspace*{0.5cm}
\begin{figure}[!hbpt]
\begin{center}
\hspace*{0.8cm}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/solid_k108_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/solid_k108_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/fluid_k108_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/fluid_k108_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/solid_k106_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/fluid_k106_large.pdf}
\\
\vspace*{0.1cm}
\hspace*{0.8cm}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/fluid_k106_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/solid_k105_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/solid_k105_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/fluid_k105_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/fluid_k105_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/comparison_large_solid.pdf}
\\
\vspace*{0.1cm}
\hspace*{0.8cm}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/comparison_close_solid.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/comparison_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/GAJO_Figures/comparison_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/solid_k_108_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/solid_k_108_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/fluid_k_108_large.pdf}
\\
\vspace*{0.1cm}
\hspace*{0.8cm}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/fluid_k_108_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/solid_k_106_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/solid_k_108_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/fluid_k_106_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/fluid_k_108_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/solid_k_105_large.pdf}
\\
\vspace*{0.1cm}
\hspace*{0.8cm}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/solid_k_105_close.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/fluid_k_105_large.pdf}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/GAJO.original/HHT/fluid_k_105_close.pdf}
%\caption{\label{Fig1}{Time histories of solid displacement due to longitudinal wave at the depth of 1 cm below the surface. Comparison between numerical results (FEM) and analytical solution by \cite{Gajo1995b} for the case of viscous coupling, $k$, equal to $10^{8} cm^3s/g$. Two different sets of Newmark parameters were used for the numerical analysis.}}
\end{center}
\end{figure}
\end{frame}
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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}
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}