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\begin{document}
%
\begin{center}
{\Large \bf On Validation of Fully Coupled Behavior of Porous Media using Centrifuge Test Results}
%{\Large \bf
%Validation Procedures for 
%Simulation of Fully Coupled Behavior of 
%Porous Media
%}
\end{center}


\vspace*{0.5cm} 

\begin{center}
{\large \bf Panagiota Tasiopoulou} \\
{PhD Student, 
National Technical University of Athens, 
Athens, Greece}\\ ~ \\ 
%%%%%%%%%%%%%%%%%%%%
{\large \bf Mahdi Taiebat} \\
Associate Professor, 
Department of Civil Engineering, The University of British Columbia, 
Vancouver\\ ~ \\
%%%%%%%%
%Mahdi Taiebat \scalebox{1}{\includegraphics[viewport=0 4.5 0 0]{mahdi.eps}}
{\large \bf Nima Tafazzoli} \\
Geotechnical Engineer, Tetra Tech EBA, Vancouver, BC, Canada \\ ~ \\ 
% %%%%%%%%
%%%%%%%%%
{\large \bf Boris Jeremi{\'c}}\footnote{Corresponding Author, Department of Civil and Environmental Engineering,
University of California, One Shields Ave, Davis, CA, 95616.
\texttt{jeremic@ucdavis.edu}} 
\\
Professor, 
Department  of  Civil  and  Environmental  Engineering,  University  of
California, Davis, CA,  
and  Faculty  Scientist,  Earth  Science  Division,  Lawrence  Berkeley National
Laboratory, Berkeley, CA.
\\
\end{center}
%%%%%%%%
%and
%%%%%%%%%

keywords: verification   and  validation,  
          finite  elements,  
          fully coupled analysis, 
          porous media

%\end{abstract}

%  
%\linenumbers


\section*{Abstract}

  
Modeling     and    simulation  of    mechanical    response  of infrastructure 
object, solids  and  structures,  relies  on the use of computational models to
foretell  the  state  of  a  physical  system  under  conditions  for which such
computational  model  has not been validated. Verification and Validation (V\&V)
procedures  are the primary means of assessing accuracy, building confidence and
credibility  in  modeling  and  computational  simulations  of behavior of those
infrastructure  objects.  
%
Validation is the process of determining a degree to which a model is
an  accurate  representation  of  the  real  world  from  the perspective of the
intended  uses  of the model. It is mainly a physics issue and provides evidence
that the correct model is solved \citep{Oberkampf2002}.


Our   primary  interest is in modeling and simulating  behavior  of
porous  particulate  media  that  is  fully saturated with pore fluid, including
cyclic mobility and liquefaction.
Fully  saturated  soils  undergoing  dynamic  shaking  fall in this
category.
%
Verification  modeling  and  simulation  of  fully  saturated  porous  soils  is
addressed  in more detail by \citep{Tasiopoulou2014}, and
in this paper we address validation.
%
A set of centrifuge experiments is used for this purpose.
Discussion is provided assessing the effects of scaling laws
on centrifuge  experiments  and  their  influence on the validation.
%
Available    validation  test  are  reviewed  in  view of first and second order
phenomena and their importance to validation. For example, dynamics behavior of the
system, following  the dynamic time, and dissipation of the pore fluid pressures, 
following diffusion time, are not happening in the same time scale and those 
discrepancies are discussed.
%
Laboratory tests, performed on soil that is used in centrifuge experiments, were
used to calibrate material models that are then used in a validation process.
%
Number  of    physical   and  numerical  examples are used for validation and to
illustrate presented discussion.
%
In particular, it is shown that for the most part, numerical prediction of behavior, using
laboratory  test  data  to  calibrate  soil material model, prior to centrifuge
experiments,  can  be  validated  using  scaled  tests.  There  are,  of course,
discrepancies, sources of which are analyzed and discussed.



 \newpage

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\section{Introduction}

Numerical predictions of behavior of civil engineering solids and structures has
gained   a  significant  popularity  in  last  decades,  particularly  with  the
availability  of fast digital computers and a number of commercial and research 
programs  (numerical modeling and simulations tools) that feature nice graphical
user  interfaces  (GUIs).  While  expansion of use of numerical prediction tools
brings  great  promises for improved design  (improved safety and economy) there
also    exists   a  danger  of  using  numerical  prediction  tools  for modeling
and  simulating  phenomena  for  which  these  tools  have not been verified and
validated. 

Verification  and
Validation    (V\&V)    procedures    are    the   primary   means  of
assessing accuracy,
building  confidence  and credibility in modeling and computational simulations.
%
Verification   is  the  process  of  determining  that  a  model  implementation
accurately  represents the developer's conceptual description and specification.
It is mainly a mathematics issue, and provides evidence that the model is solved
correctly. 
%
Validation is the process of determining a degree to which a model is
an  accurate  representation  of  the  real  world  from  the perspective of the
intended  uses  of the model. It is mainly a physics issue and provides evidence
that the correct model is solved \citep{Oberkampf2002}.


Verification      and      validation      has    recently   gained   increased 
attention,  with the understanding that numerical prediction results can only be
trusted if proper verification and validation has been performed
\citep{Mroz88, Arulanandan93, Zienkiewicz94, Roach1998, 
Oberkampf2002,     Oden2004,     Babuska2004,    Oden2010a,    Oden2010b,
Oberkampf2010,  Bielak2010,  Roy2011}.



In  this paper, we address the issue of validating the modeling of fully coupled
behavior  of  particular  materials  (granular  soil)  using scaled models under
increased   gravity, so called centrifuge   modeling. 
%
Basics of validation procedures are presented in Section~\ref{VandV_P}.
Detailed discussion on scaling laws, as they apply to our examples, centrifuge tests, and
validation results with comments are presented  in  Section~\ref{VCBM}.
%
Details          of        the        u-p-U    formulation    are    given    by
\citet{Tasiopoulou2014}  and  will not be repeated here. In addition, validation
of  the elastic-plastic material model used was presented by~\cite{Jeremic2007e}
and will not be covered here either.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Validation Procedures}
\label{VandV_P}


Validation  procedures  are used to provide evidence that numerical analyst have
chosen  the right models for modeling phenomena in question. As such, validation
procedures  are  tightly  coupled  to  the  physics  (mechanics) of the problem.
Validation  procedures  give  us  information  about  how  much we can trust the
numerical simulation results.


The role of validation is graphically shown in Figure~\ref{roleVandV}. 
%   
\begin{figure}[!htb]
\begin{center}
%{\includegraphics[width=10.0cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/RoleVV_NEW_knowledge.pdf}
\includegraphics[width=7.0cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/RoleVV_NEW_knowledge.eps}
\caption{Role  of Verification and Validation in relation to the knowledge about
reality (graphics inspired by \citet{Oberkampf2002}).}
\label{roleVandV}
\end{center}
\end{figure}
%
It  is  important to note that both verification and validation procedures are necessary
in order to gain confidence in numerical modeling results, and to be able to make 
informed  decisions  about  the  behavior  of  a problem being analyzed. It is also
important to note that real physical behavior of a mechanical system is never fully known. This is
a result of
a macro scale
interpretation   of   the   Heisenberg   uncertainty  principle
\citep{Heisenberg1927},  stating that one cannot obtain 
position and momentum of a material  particles at the same time
resulting  from  some  deterministic or stochastic loading.
%
Hence,             only  an approximate knowledge (with some level of
certainty)  of  behavior  of  an object (solid and/or structure), can be gained.
This  argument emphasizes the need for stochastic treatment of both physical and
numerical modeling and simulations \citep{Oberkampf2002}.

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\paragraph{Independence of Computational Confidence Assessment}
%\begin{enumerate}
%   \item V\&V conducted by the computational tool developer,  {\it No Independence}
%   \item V\&V conducted by a user from same organization
%   \item V\&V conducted by a computational tool evaluator contracted by developer's organization
%   \item V\&V conducted by  a computational tool evaluator contracted by the customer
%   \item V\&V conducted by  a computational tool evaluator contracted by the a legal authority  {\it High Independence}
%\end{enumerate}
  


A more detailed analysis of validation  reveals the importance
of   a  hierarchy  of  experimental  data.  Figure~\ref{VerifValidFund01}  shows
relationship  of  real  world  behavior  with  verification  and  (emphasis on)
validation.
%-- 
\begin{figure}[!htb]
\begin{center}
{\includegraphics[width=11cm]{/home/jeremic/tex/works/Papers/2011/Validation_Coupled_Systems/VerifValidFund01.eps}}
\caption{Relationship   of  verification  and validation to the real world, with
emphasis on validation and experimental data (inspired by \citet{Oberkampf2002}).}
\label{VerifValidFund01}
\end{center}
\end{figure}
%
As  noted by \citet{Oberkampf2002}, validation can be understood as a process to
determine how accurately the model (focusing on its intended use) represents the real world, 
%
The  validation experiments\footnote{As opposed to traditional experiments which
are  used  to  improve  (a) understanding of physics and (b) mathematical models
of/for  a  phenomena  in  question.}  used  for  this  purpose  are designed and
performed  to  estimate  computational model's ability to model
defined  physical  behavior/phenomena.  In  a sense, the numerical modeling tool
(computational simulation tool) becomes the main customer of designed validation
experiment.
%
Ideally,  a  validation  experiments should be jointly designed and performed by
physical modeler (experimentalist)  and numerical modeler (numerical analyst).
Validation  experiment  should  be  able  to capture relevant/important physics,
where  physical  effects  of  primary  importance  are  properly  modeled  while
secondary effects might be modeled using some level of approximation.


It is important to note that the validation domain is almost always exclusive of
the  application  domain.  That is, real physical phenomena that we are interested in,
cannot  be  fully  physically  modeled  due  to  complexity,  cost,  size,  etc.
%
For example,
civil  engineering  systems  like  bridges,  buildings,  port  facilities, dams,
nuclear  power plants, etc. are to complex, expensive and large to be tested for
all loading scenarios of interest. 
%
Even if the engineering system is small (less expensive, complex), environmental
influences  (generalized  loads,  conditions,  wear  and tear) are hard to model
physically.
%
Validation  domain  thus  represents  a  simplification  of  application domain. 
Figure~\ref{VPI03} shows a relationship of validation and application domain for
civil  engineering  applications (structural and soil mechanics) that are almost
always  exclusively  non-overlapping  in  their  systems  parameters  and system
complexity.  
%
\begin{figure}[!htb]
\begin{center}
{\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Presentation/2004/VandV/VPI03.eps}}
\caption{Relationship of the    validation  domain to the application domain, which, in general
for     civil  engineering  application  (structural  and  soil  mechanics)  are
exclusively non-overlapping \citep{Oberkampf2002}).}
\label{VPI03}
\end{center}
\end{figure}
%
The  inference  from  validation  to  application  domain  is  done numerically.
Such inference is based on physics while uncertainties in material behavior, loads,
geometry, etc. have to be addressed as well.
%
The  importance  of  uncertainty  quantification  in  experiments  and numerical
predictions  cannot  be  overstated.  All  relevant  sources  of  uncertainty in
physical models need to be
identified  and  uncertainties  estimated.  Those  uncertainties then need to be
propagated through modeling and simulation process.




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






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Validation of Coupled Behavior Modeling}
\label{VCBM}
















%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Scaling Laws}
\label{Discussion of Scaling laws}

%\vspace{10 mm}

Scaling  laws  are  of  great  importance,  not only for the centrifuge modeling
itself,  but  also  for  the  accurate  numerical reproduction of the centrifuge
tests.  
The important scaling laws for higher gravity modeling of liquefaction are
concerning the dynamic time and the permeability and consequently the diffusion time. 
%
The        Darcy permeability of the centrifuge model (under increased gravity field of 
$N\times g$) is $N$ times larger than permeability that was measured
in  the  laboratory (under gravity field of $1\times g$).  
%
This  leads  to a difference between the scaling factors for  the  dynamic  time
and  the  diffusion  time if the same materials (water and soil) are used in the
model  and  prototype.  This conflict in time scales is essential to scaling the
centrifuge  measurements  up  to  the  prototype  scale since both generation of
excess  pore fluid pressures (dynamic time) and dissipation of excess pore fluid
pressure  (diffusion  time)  happen at the same time throughout shaking and are
equally important to the modeling (physical and/or numerical) of liquefaction.
%


In  order  to  analyze  the  appropriate  scaling  of the permeability and the
diffusion time, we consider three different cases: A, B and C, as illustrated in
Figure~\ref{scale}. It is assumed that water fills the voids of the soil in all
cases. 

\begin{itemize}

  \item[] Case   A  is  the  object   in prototype (original) scale representing 
  the original soil conditions, 

  \item[] Case B is the centrifuge model in
model  scale,  and 

  \item[] Case C  is the scaled up prototype model derived  from  scaling  the centrifuge model up to
prototype scale. 

\end{itemize}

\begin{figure}[!hbt]
%\begin{Large}
%\begin{sffamily}
\begin{center}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/scale.eps}
\end{center}
\caption{\label{scale}{Schematic  illustration  of the centrifuge 
modeling concepts and  scaling  laws  described  in detail in Section \ref{Discussion of
Scaling  laws}.  Case  A represents the object for simulation, Case B represents
the  centrifuge  model and Case C represents what has been actually simulated in
prototype  scale.  The  symbols,  $k_D$,  $K$  and  $T$,  correspond  to Darcy's
permeability, specific permeability and consolidation time respectively.}}
%\end{sffamily}
%\end{Large}
\end{figure}


In literature, the term "prototype model" is sometimes confusingly used 
to characterize either Case A or Case C,
without  further  clarifications.  Herein,  Case A is characterized as "Original
Model"  and  Case  C as "Prototype Model". Scaling factors for all the important
quantities  related  to  our study are presented in Table \ref{Scaling} \citep{Wood2004}.  
%
\begin{table}[!htb]
\caption{ \label{Scaling}
Scaling factors for a case of water filling the pore space (voids) in the soil.}
\begin{center}
%\begin{spacing}{1.4}
\begin{tabular}{c|c|c|c}
                                 & \bf{Case A}           & \bf{Case B}           & \bf{Case C}           \\
\hline
                                 & Original Model   & Centrifuge Model &  Prototype Model  \\                      
 quantity                        & 1g               & Ng               & 1g                  \\
                                 &                  & Numerical Model 1               & Numerical Model 2                  \\
\hline
     length                      & N                & 1                & N                    \\
     mass density                & 1                & 1                & 1         \\
     stress                      & 1                & 1                & 1   \\
     strain                      & 1                & 1                & 1  \\
     displacement                & N                & 1                & N  \\
     acceleration                & $1/N$            & 1                & $1/N$  \\
     Darcy's permeability        & $1/N$            & 1                & 1  \\
     specific permeability       & 1                & 1                & N  \\
     time (diffusion)            & $N^2$            & 1                & N   \\
     time (dynamic)              & N                & 1                & N  \\
     frequency                   & $1/N$            & 1                & $1/N$  \\
\end{tabular}
%\end{spacing}
\end{center}
\end{table}





The  base, to  which  all the scaling factors refer to, in Table
\ref{Scaling},  is  the  model  scale. The column corresponding to the "Original
Model"  presents  the  scaling factors needed to be applied to the quantities in
model  scale  in  order to reproduce what has been intended to be simulated from
the  beginning - the "Original Model". The conflict in time scales is related to
the "Original Model". The column corresponding to the "Prototype Model" presents
the  scaling  factors  needed  to be applied to the quantities in model scale in
order to reproduce what has been actually simulated in prototype scale -
the  "Prototype  Model"  .  Thus,  comparison between what  was intended to be
studied (Case A) and what was studied in reality (Case C) can reveal scaling problems.
A detailed analysis of scaling for each of three cases (A, B and C) follows.

%\vspace{10 mm}

\paragraph{Case  A}  is the object for simulation and represents the prototype soil
conditions  and  properties that are intended to be simulated through centrifuge
modeling. Darcy's permeability, $k_{DA}$, of this type of soil has been measured
in  the lab. Case A could be simply described as a model N times larger than the
centrifuge  model  with the same soil under 1g. The specific permeability, $K_A$
and  the  time needed for completion of the 1D consolidation process, $T_A$, can
be  estimated  using  $k_{DA}$  through  Equation  \ref{case_A_k}  and Equation
\ref{case_A_T}  respectively.


\begin{eqnarray}
K_A=\frac{k_{DA}}{\rho_f\times g}
\label{case_A_k}
\end{eqnarray}



\begin{eqnarray}
T_A=\frac{H^2}{C_v}=\frac{H^2\times \rho_f \times g}{k_{DA}\times E_{eod}}=\frac{H^2}{K_A\times E_{oed}}
\label{case_A_T}
\end{eqnarray}
%
Here $\rho_f$ is the mass density of the fluid
(water),  $g$ is the acceleration of gravity ($9.81$~$\rm{ m/s^2}$) in this case,
$H$  is  the thickness of the soil layer in real/original scale and $E_{oed}$ is
the one dimensional soil stiffness:


\paragraph{Case  B} represents the centrifuge model which consists of the same type
of  soil  as  in  case A  with  the  same  relative  density.  However, Darcy's
permeability  is  proportional  to  the  gravity, as it is indicated by Equation
\ref{case_A_k}.  
%
The centrifuge gravity field (model) is $N\times g$, 
and  the  specific  permeability,  $K$,  is a soil constant
(independent   of  the  permeant). It follows that by  neglecting  the changes
of  void  ratio  and  the  gravity  level, the actual  Darcy's 
permeability of the
centrifuge  model  is N times larger that that of
Case      A,      as    shown    in  Equation  \ref{case_B_k} \citep{Wood2004}. 
%
\begin{eqnarray}
k_{DB}=K_B\times \rho_f\times N\times g=K_A\times \rho_f \times N\times g=N\times k_{DA}
\label{case_B_k}
\end{eqnarray}
%
Furthermore, the actual
consolidation time of Case B becomes $N^2$ smaller than that of Case A, as shown
in  Equation \ref{case_B_T}. 
%
\begin{eqnarray}
T_B=\frac{(H/N)^2}{K_B\times E_{oed}}=\frac{H^2}{N^2\times K_A\times E_{oed}}=\frac{T_A}{N^2}
\label{case_B_T}
\end{eqnarray}
%
In other words, the appropriate time scaling factor
for the centrifuge measurements for diffusion is $N^2$,  in order to get a
similar  response with Case A. On the other hand, the appropriate scaling factor
for  the  dynamic  time  is $N$. This fact leads to a problem  with realistic simulation
of  the    Case    A using centrifuge model. This stems from the fact that it is
difficult  to  separate  the dynamic and the diffusion times since they are both
contributing to first rate physical effects and cannot be separated (both are of
the same importance for liquefaction and cyclic mobility phenomena).




%\vspace{10 mm}

\paragraph{Case  C}  represents  what  has  been  actually  simulated and tested
in  (scaled up to) prototype scale, through the centrifuge modeling process. The
centrifuge model under a
gravity  field of $N\times g$ corresponds to a soil layer N times larger in size
in  prototype  scale,  so that the stress field is common in both models. Case C
can  be  simply  described  as  a model N times larger than the centrifuge model
consisting  of soil with the same Darcy permeability (N times less than that in
Case  A).  Darcy's permeability of the prototype model (Case C) is equal to that
of  the  centrifuge model (Case B), which means N times larger than that in Case
A.  Theoretically,  the  fact  that $k_{DB}=k_{DC}$ leads to different values of
specific  permeability  between  the  centrifuge model and the prototype one. In
detail,  the specific permeability of Case C, $K_C$, is N times larger than that
of  Case  B  and  Case  C,  as described by Equation \ref{case C_k}. 
%
\begin{eqnarray}
K_C=\frac{k_{DC}}{\rho_f\times g}=\frac{k_{DB}}{\rho_f\times g}=N\times K_B=N\times K_A
\label{case C_k}
\end{eqnarray}
%
This is the
recommended  value of the specific permeability that should be used in numerical
simulation  of  the centrifuge tests in prototype scale. Moreover, the diffusion
time of the prototype model is estimated (Equation \ref{case C_T}) to be N times
less  than that of the centrifuge model, which coincides with the scaling factor
for  the  dynamic  time. 
%
\begin{eqnarray}
T_C=\frac{H^2}{K_C\times E_{oed}}=\frac{H^2}{N\times K_B\times E_{oed}}=\frac{T_B}{N}=N\times T_A
\label{case C_T}
\end{eqnarray}
%
However, the diffusion time of the prototype model is N
times larger when compared  to Case A.



Since the permeability is proportional to gravity, this presents a problem for
modeling real scale case (Case A) using  changed (larger) gravity field (Case B),
if water is used as pore fluid.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Use of Higher Viscosity Replacements Fluids.}
%


In order to overcome this scaling discrepancy and achieve the same value of
permeability  in  all cases, fluids with larger viscosity are chosen to fill the
voids  within the soil, according to Equation \ref{1000} below:
%
\begin{eqnarray}
k_D=\frac{K\times \rho_f g}{\mu}
\label{1000}
\end{eqnarray}
%
where $\mu$ is pore fluid viscosity.  
%
The  Equations  \ref{v_A} and \ref{v_B} 
%
\begin{eqnarray}
k_{DA}=K\times \rho_w g
\label{v_A}
\end{eqnarray}
%
\begin{eqnarray}
k_{DB}=\frac{K\times \rho_f\times N\times g}{\mu}
\label{v_B}
\end{eqnarray}
%
indicate that the pore fluid
viscosity should be equal to $\rho_f\times N/\rho_w$, where $\rho_w$ is the mass
density  of the water, so that $k_{DA}=k_{DB}$. In that case, the scaling factor
for the diffusion time is equal to $\rho_f\times N/\rho_w$ instead of $N^2$ when
water    is  used. In particular, if the pore fluid has the same mass density as
water, then the time scaling factor for diffusion is the same as the one for
dynamic events.


%\vspace{10 mm}

\cite{Wood2004}  suggests  the   use  of silicone fluid as a replacement fluid.
Another  possibility  is  to  use a solution of Hydroxypropyl methylcellulose in
water,  which,  when mixed in right proportion, increases the viscosity of water
from  $1 mm^2/s$ to approximately $25 mm^2/s$ \citep{Kulasingam2004}. Increasing
viscosity  more  than  that amplifies the problem of proper (full) saturation of
the sample, which than significantly affects other aspects of the experiment. By
using  scaled  up viscosity of (up to) $\mu=25 mm^2/s$, the geometric scaling of
model  is  also  limited to $25$. That means that any centrifuge experiment that
uses $\mu=25 mm^2/s$ and is modeled in a centrifuge spinning at 50g level, using
original  soil,  will  have a scaling factor for the diffusion time equal to 
$25({1}/{50})^2/{1}   =   0.01 =1/100  $   and   for   the   dynamic   time   equal   to
$({1}/{50})\sqrt{{1}/{1}}  = 0.02 = 1/50$. That is, the geometric scaling will properly
match  the dynamic time scale while diffusion time scale will be half of what it
is  supposed  to be. This means that the diffusion is occurring twice as fast in
comparison  with  the  original  model.  Similarity  of  the  original  and  the
centrifuge  model  does not exist in this case. The larger the geometric scaling
is,  when compared  to  the fixed value of viscous scaling (usually not more than
25), the larger the
discrepancy  in  time  scale  is.  This inconsistency in scaling creates the so
called  distorted  models  as  they inappropriately scale one of the first order
phenomena.








%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Numerical Simulation of RPI Centrifuge Test (Model No1, Test 2) by Taboada and Dobry (1993)}
\label{Numerical Simulation of RPI centrifuge test (Model No1, Test 2) by Taboada and Dobry (1993)}





Numerical modeling and  simulation of the liquefaction is
performed   using  the  $u-p-U$ \citep{Zienkiewicz84, Jeremic2007e} formulation 
in combination with the
constitutive  model  by  \cite{Dafalias2004}. For validation, results
from  the centrifuge tests of Model No. 1 (test 2) presented by \cite{Dobry1993}
from VELACS project are used.










A    schematic    configuration   of the centrifuge model No.1 is illustrated in
Figure~\ref{conf_cent}. The soil consists of a uniform layer of Nevada sand with
relative  density  $D_r\approx40\%$  and is fully-saturated with the pore fluid.
The thickness of the soil layer is 20 cm in model scale and the field of gravity
applied  to  the model is 50g. The input motion applied to the base of the model
is also shown in Figure~\ref{conf_cent} in real time (model) scale. According to
the  scaling laws provided by \cite{Wood2004}, the prototype model is 10 m thick
and    its  Darcy  permeability is 50 times greater than the value obtained from
lab  tests, as reported by \cite{Arulmoli1992}. Accelerations, displacements and
pore pressures
were  measured during testing at select locations. The locations and the type of
measurements recorded are shown in Table \ref{Location_of_Instrumentation}.


\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=10cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/conf_cent.ps}
\end{center}
\caption{\label{conf_cent}{Schematic  configuration  of Model No. 1, Test 2, RPI
\citep{Dobry1993}.}}
\end{sffamily}
\end{Large}
\end{figure}


\begin{table}[!htb]
\caption{ \label{Location_of_Instrumentation}
Location and Type of Measurement in Model No. 1, Test 2, RPI }
\begin{center}
\begin{spacing}{1.4}
\begin{tabular}{c|c|c|c}
      \bf              &                   & {Depth in}     & {Depth in}              \\
          Measurement  & {Instrument ID}   & {model scale}  & {prototype scale}               \\
\hline
      horizontal acceleration     & $AH 1$     & $20 cm$   & $10 m$       \\
      horizontal acceleration     & $AH 3$     & $0 cm$    & $0 m$       \\
      horizontal acceleration     & $AH 4$     & $5.2 cm$  & $2.6 m$       \\
      horizontal acceleration     & $AH 5$     & $10 cm$   & $5 m$       \\
      horizontal displacement     & $LVDT 3$   & $0.9 cm$  & $0.45 m$        \\
      horizontal displacement     & $LVDT 4$   & $5 cm$    & $2.5 m$        \\
      horizontal displacement     & $LVDT 5$   & $10 cm$    & $5 m$        \\
      horizontal displacement     & $LVDT 6$   & $15 cm$   & $7.5 m$        \\
      vertical displacement       & $LVDT 1$   & $0 cm$    & $0 m$        \\
      pore fluid pressures              & $P 1$      & $2.9 cm$  & $1.45 m$        \\
      pore fluid pressures              & $P 2$      & $5.2 cm$  & $2.6 m$        \\
      pore fluid pressures              & $P 3$      & $10 cm$    & $5 m$        \\
      pore fluid pressures              & $P 4$      & $15 cm$    & $7.5 m$        \\
\end{tabular}
\end{spacing}
\end{center}
\end{table}



%\vspace{10 mm}

Prior   to   the   numerical  simulation,  calibration  of SANISAND  \citep{Dafalias2004}
constitutive  model  was  performed for Nevada sand, using soil data reported by
\cite{Arulmoli1992}  in  the  framework  of VELACS project. \cite{Taiebat2009a}
present the values of the material parameters of SANISAND for Nevada sand, shown
in  Table  \ref{DM04}.  Fig.  \ref{nevada test} compares the
stress  paths  and  stress-strain  loops  for  Nevada sand with $D_r\approx40\%$
obtained from lab tests with those obtained from calibration process.

%\vspace{10 mm}

\begin{table}[tbh]
\caption{ \label{DM04} Material parameters of Dafalias-Manzari model. }
\begin{center}
\begin{spacing}{1.4}
\begin{tabular}
[c]{l|c|c||l|c|c}
\multicolumn{2}{ l|}{\bf{Material Parameter}} & \bf Value &
\multicolumn{2}{|l|}{\bf{Material Parameter}} & \bf Value\\
\hline
Elasticity & $G_{0}$ & 150 kPa & Plastic modulus & $h_{0}$ & 9.7\\
& $v$ & 0.05 &  & $c_{h}$ & 1.03\\
Critical sate & $M$ & 1.14 &  & $n_{b}$ & 2.56\\
& $c$ & 0.78 & Dilatancy & $A_{0}$ & 0.81\\
& $\lambda_{c}$ & 0.027 &  & $n_{d}$ & 1.05\\
& $\xi$ & 0.45 & Fabric-dilatancy & $z_{max}$ & 5.0\\
& $e_{r}$ & 0.83 &     & $c_{z}$ & 800.0\\
Yield surface & $m$ & 0.05 &  &  & \\
\end{tabular}
\end{spacing}
\end{center}
\end{table}


%%\vspace{10 mm}


%======================
\begin{figure}[!htb]
\begin{center}
{\normalsize 
\mbox{
\subfigure{Experiment} \quad 
\hspace{4cm}
\subfigure{Simulation}
} \\ 
\setcounter{subfigure}{0}%
\mbox{
\subfigure[]{\includegraphics[width=0.48\textwidth]{CIUC-Cyclic-Nevada-test-1.eps}} \quad 
\hspace{-1cm}
\subfigure[]{\includegraphics[width=0.48\textwidth]{CIUC-Cyclic-Nevada-model-1.eps}}
} \\ 
\mbox{
\subfigure[]{\includegraphics[width=0.48\textwidth]{CIUC-Cyclic-Nevada-test-2.eps}} \quad 
\hspace{-1cm}
\subfigure[]{\includegraphics[width=0.48\textwidth]{CIUC-Cyclic-Nevada-model-2.eps}}
}\\
}
\end{center}
\caption{
\label{nevada test}
{Simulations  versus experiments in undrained triaxial tests on Nevada sand with
$D_r\approx40\%$} published by \cite{Taiebat2009a}. The experimental data can be
found in \cite{Arulmoli1992}.}
\end{figure}
%======================

%
%\begin{figure}[!hbt]
%\begin{Large}
%\begin{sffamily}
%\begin{center}
%\includegraphics[width=16cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/nevada_test.ps}
%\end{center}
%\caption{
%\label{nevada test}
%{Simulations  versus experiments in undrained triaxial tests on Nevada sand with
%$D_r\approx40\%$} published by \cite{Taiebat2009a}. The experimental data can be
%found in \cite{Arulmoli1992}.}
%\end{sffamily}
%\end{Large}
%\end{figure}

%\vspace{10 mm}

 After  the  calibration of the model, numerical simulation of direct shear test
 was  conducted  so as to find the relationship between the cyclic resistance of
 Nevada  sand  with  relative  density $D_r=40\%$, versus the number of cycles to
 liquefaction, and compare it with the experimental data by \cite{Arulmoli1992}.
 The  numerical  model  consisted of a single $u-p-U$ brick element subjected to
 cyclic  horizontal  shear  loading  under  undrained  conditions. Two different
 initial  vertical effective stresses were considered, $20$kPa and $70$kPa, in
  order    to    investigate    the   effect  of the initial confining stress, as
  captured by the model. The cyclic stress
 ratio,  CSR,  is defined as the ratio of horizontal shear stress to the initial vertical
 effective  stress.  The number of cycles to liquefaction both in the experiment
 and  the analysis is determined according to the following criteria: (i) either
 the  axial strain exceeds 1.5\% or (ii) the excess pore water pressure ratio
 becomes  equal  to one. Figure~\ref{CSR} illustrates the comparison between the
 numerical  and experimental data, showing, a fairly satisfactory agreement.
%, although the effect of
%the  vertical effective  stress on number of cycles cannot be properly captured by the model
% pis negligible after normalizing the shear stress to
% CSR.

%\vspace{10 mm}

\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=12cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/CSR.ps}
\end{center}
\caption{\label{CSR}{Cyclic  shear  stress  ratio  for Nevada sand with relative
density  $D_r=40-45\%$  versus  the number of cycles to liquefaction. Comparison
between  the  numerical  results  using material model from \cite{Dafalias2004} and the experimental
data by \cite{Arulmoli1992}.}}
\end{sffamily}
\end{Large}
\end{figure}

%\vspace{10 mm}


Most  frequently,  the  numerical simulation of centrifuge tests is conducted in
prototype scale. However, due to conflicts related to the appropriate scaling of
dynamic   time   and   dissipation-time,   when   water   is   used   in   tests
(\cite{Wood2004}),  a  different  approach  was  adopted  in the present study.
%
%Details  on  the  scaling  factors  are  presented in Section \ref{Discussion of
%Scaling laws}. 
Two different numerical models were used:

%\vspace{10 mm}

\begin{enumerate}

  \item Numerical Model \#1 in model scale.
  
  The   numerical  model \#1 consists  of  a  soil  column  of twenty 8-node $u-p-U$
  brick  elements  with dimensions $1cm \times 1cm \times 1cm$. The total height
  of the soil column is 20 cm. The gravity applied to the model is equal to 50g.
  The  input motion is shown in Fig. \ref{conf_cent}. The specific permeability,
  $K$,   is   $3.2\times10^{-6}  cm^3s/g$.  The  results  of  the  analysis  are
  illustrated  in  Fig.  \ref{disp_model}  and \ref{acc_model} in model scale in
  terms of time histories of displacements and accelerations respectively.
  
  
\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/model/disp_model.ps}
\end{center}
\caption{\label{disp_model}{Time histories of horizontal displacements at the depths of 0, 5 ,10 and 20 cm from the surface in model scale, as obtained from Numerical Model 1.}}
\end{sffamily}
\end{Large}
\end{figure}

\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/model/acc_model.ps}
\end{center}
\caption{\label{acc_model}{Time histories of horizontal acceleration at the depths of 0, 5 ,10 and 20 cm from the surface in model scale, as obtained from Numerical Model 1.}}
\end{sffamily}
\end{Large}
\end{figure}





  \item  Numerical Model \#2 in prototype scale.
  
  The    numerical  model \#2 consists of a soil column of twenty 8-node u-p-U brick
  elements  with  dimensions  $0.5m\times0.5m\times0.5m$. The total thickness of
  the  soil  column  is 10 m. The gravity applied to the model is equal to $1$g.
  The  input  motion  has been modified to prototype scale (the dynamic time has
  been  multiplied  by  50  and  the  acceleration  has been divided by 50). The
  specific  permeability, $K$, is $1.6\times10^{-4} cm^3s/g$. The results of the
  analysis  are  illustrated  in  Figs.  \ref{disp_pm_comp} to \ref{pe_pm_comp}.
  Results  are  presented  in prototype scale and are comparing Numerical Models
  \#1  and  \#2  after  scaling laws between Case B and C of Table~\ref{Scaling}
  were  applied  - apart from strain and stresses which are in the same scale by
  default for both models.


\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/proto_model_comp/disp_proto_model_comp.ps}
\end{center}
\caption{\label{disp_pm_comp}{Time  histories  of horizontal displacement at the
depths  0,  2.5,  5,  7.5  m  from  the  surface. Comparison between the results
obtained from Numerical Model 1 and Numerical Model 2, in prototype scale.}}
\end{sffamily}
\end{Large}
\end{figure}



\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/proto_model_comp/acc_proto_model_comp.ps}
\end{center}
\caption{\label{acc_pm_comp}{Time  histories  of  horizontal acceleration at the
depths 0, 2.5, 5, 10 m from the surface. Comparison between the results obtained
from Numerical Model 1 and Numerical Model 2, in prototype scale.}}
\end{sffamily}
\end{Large}
\end{figure}

\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/proto_model_comp/pp_proto_model_comp.ps}
\end{center}
\caption{\label{pp_pm_comp}{Time  histories of excess pore water pressure at the
depths  1.5,  5,  7.5,  9  m  from  the  surface. Comparison between the results
obtained from Numerical Model 1 and Numerical Model 2, in prototype scale.}}
\end{sffamily}
\end{Large}
\end{figure}

\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/proto_model_comp/settle_proto_model_comp.ps}
\end{center}
\caption{\label{settle_pm_comp}{Time  histories  of vertical displacement at the
depths  0,  3,  6, 9 m from the surface. Comparison between the vertical outward
movement of the water and the settlement of the soil, in prototype scale.}}
\end{sffamily}
\end{Large}
\end{figure}

\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/proto_model_comp/pq_proto_model_comp.ps}
\end{center}
\caption{\label{pq_pm_comp}{Horizontal  shear  stress  versus effective vertical
stress  at  the depths of 1.25, 5, 7.5, 9 m. Comparison between the stress paths
obtained from Numerical model 1 and Numerical Model 2.}}
\end{sffamily}
\end{Large}
\end{figure}

\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/proto_model_comp/pe_proto_model_comp.ps}
\end{center}
\caption{\label{pe_pm_comp}{Shear  stress-strain loops at the depths of 1.25, 5,
7.5,  9  m.  Comparison between the stress paths obtained from Numerical model 1
and Numerical Model 2.}}
\end{sffamily}
\end{Large}
\end{figure}



\end{enumerate}


The  soil  properties  used  for  the  both  numerical models are shown in Table
\ref{Soil_Properties_for_centrifuge_test}. The Newmark time integration method is used 
since it was shown in verification study \citep{Tasiopoulou2014} to better dissipate  high
frequencies      introduced      in      the  coupled system  by the discretization
process. The Newmark parameters
used  for  all  the  numerical  analysis  for  both  models  are:  $\gamma=0.7$,
$\beta=0.42$.    The    column  is horizontally excited during a second stage of
loading, after first stage self-weight loading. It
should  be  noted that the self weight loading is performed on an initially zero
stress    (unloaded)  soil  column  and that the material model and constitutive
integration algorithm is versatile enough to
follow  through this early loading with proper parameter evolution. The boundary
conditions  are  such that the soil and water displacement degrees at the bottom
surface  are  fixed,  the  pore fluid pressure  degrees  are free; the soil and water
displacement  degrees  at  the upper surface are free, however the pore pressure
degrees  are fixed (zero pore fluid pressure) to simulate the upward drainage. In order to simulate one
dimensional  shaking (shear box), all the degrees of freedom at the same level are constrained
in a master-slave fashion.

%\vspace{10 mm}

\begin{table}[!htb]
\caption{ \label{Soil_Properties_for_centrifuge_test}
Soil Properties for the centrifuge test, Model No. 1, test 2, RPI}
\begin{center}
\begin{spacing}{1.4}
\begin{tabular}{c|c|c}
       Parameter                    & Symbol    &  Value                         \\
\hline
      gravity acceleration         & $g$       & 9.81~$m/s^2$         \\
      solid particle density       & $\rho_s$  & $2.65\times 10^3$~$kg/m^3$        \\
      water density                & $\rho_f$  & $1.0\times 10^3$~$kg/m^3$        \\
      solid particle bulk modulus  & $K_s$     & $36.0\times 10^6$~$kN/m^2$              \\
      fluid bulk modulus           & $K_f$     & $2.2\times 10^6$~$kN/m^2$              \\
      porosity                     & $n$       &  0.4253                             \\
      initial void ratio           & $e_0$       &  0.74                             \\
      Darcy permeability           & $k_D$       &  $1.6\times\ 10^{-3}$m/s            \\
      Biot coefficient             & $\alpha$  &  $1.0$                    \\
\end{tabular}
\end{spacing}
\end{center}
\end{table}


The Darcy permeability of Nevada sand with $D_r\approx40\%$, varies in the
range  of  $k_D = 2.1\times10^{-5}  m/s$  to  $k_D = 3.3\times10^{-5}  m/s$,  as reported by
\cite{Arulmoli1992}.  The prototype model is considered to have 50 times larger
Darcy permeability that the one measured in the lab. The Darcy permeability used
in  the  present  study  for  both  Numerical  Models \#1 and \#2 is $1.6\times10^{-3} m/s$.
It is noted that the input parameter in our numerical code is the specific permeability,
$K$, given by Equation \ref{kspec}:

\begin{eqnarray}
 k=\frac{k_D}{\rho_f\times{G}}     
 \label{kspec}
 \end{eqnarray}
%
where $G$ is equal to $50$g for Numerical Model \#1 and $1$g for Numerical Model \#2.


%\vspace{10 mm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Discussion of Numerical Results}
\label{Conclusions}



%\vspace{10 mm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Comparison between Numerical Model \#1 and Numerical Model \#2.}
\label{numerical models}


The   results   obtained   from   the  Numerical  Model  \#1  are  shown  in  Figures
\ref{disp_model}  to  \ref{acc_model}  in model scale in order to illustrate that
this type of  analysis, using  very small size of elements and time step can be
easily performed  by the numerical code and also to   illustrate/present results that
are obtained  in model scale in a centrifuge experiment.
The numerical
results  obtained  from  Numerical  Model  \#1 were scaled to prototype scale 
according  to the scaling laws (Table, \ref{Scaling} between Column 2
and 3), so that they can compared with those of Numerical Model \#2.



This    comparison    between  Numerical  Models  \#1  and \#2, illustrated  in 
Figures~\ref{disp_pm_comp} to \ref{pe_pm_comp}, is used to  verify  the  scaling
factors derived in section
\ref{Discussion of Scaling laws}, while also giving a better insight in
the  intrinsic  differences that occur due to the difference in scale of the
two numerical models.



Numerical  results  between the two models compare well
for displacements,  accelerations  and  excess  pore water
pressure.  In  particular,  Figure~\ref{pp_pm_comp}
shows a complete agreement of the numerical results in the dissipation of the excess
pore    pressure  during and after  the  end  of  shaking. Here the excess pore
water  pressure,  $r_u$, is defined as the ratio of excess pore water pressure over
the initial
vertical effective stress.
%This  confirms the
%correctness  of  the  scaling  factors applied to the numerical analysis.
The slight difference in the results, shown in time  histories  of  displacement
 (Figure~\ref{disp_pm_comp}) is resulting from differences in the stress
paths   and   stress-strain   loops   shown   in  Figures~\ref{pq_pm_comp}  and
\ref{pe_pm_comp}  respectively. 
Numerical Model \#1 demonstrates a more
dilative behavior giving larger negative strains than Numerical Model \#2.
%
These  differences are  related to the velocity proportional
damping  associated  with  the $u-p-U$ formulation. 
Velocity  proportional damping  is related to permeability \citep{Jeremic2007e},
and    in    this    particular    case,    it    is    related   to   intrinsic
permeability which does not scale. 
As discussed by  \citet{Tasiopoulou2014},  intrinsic
(isotropic)  permeability  ${k}$  has  dimensions  of  [$L^{3}TM^{-1}$]  and  is
different from Darcy permeability (hydraulic conductivity), ($k_{D}$), which has
the  dimension of velocity, i.e. [$LT^{-1}$]. They are related by $k = k_{D} / g
\rho_f$, where $g$ is the gravitational acceleration and $\rho_f$ is the density
of  the  pore  fluid  which  is slightly changed here.
%
However,
effects  are  small  and  are  mostly apparent at lower levels of the centrifuge
model.  It  should  be  noted  that there are two types of energy dissipation in
fully-coupled systems: (i) velocity proportional damping due to viscous coupling
and  (ii)  displacement  proportional  damping  due  to  frictional  damping and
elasto-plasticity  \citep{local-87}.  In  this  case  displacement  proportional
damping  (energy  dissipation)  related to elasto-plasticity is much larger than
velocity proportional damping, and controls the response.
%OVDE




 
Figure~\ref{pq_pm_comp},  shows  the reduction  of the vertical effective stress
as  a  function of cyclic shear stress. It is also noted that vertical effective
stress rebounds after shaking stops.
%
The $r_u$ time
histories in Figure~\ref{pp_pm_comp} give clearer effective stress reduction and
rebounding. 
% Here the excess pore water pressure, $r_u$, is defined as the ratio
%of  excess pore water pressure over the initial vertical effective stress. 
We define that liquefaction occurred for values of $r_u \ge 0.90$.
%
%The  calculated  maximum  horizontal  shear strain can reach up to
%$3.2\%$.
%  
It  is also apparent  (Figure~\ref{pe_pm_comp})  that  the shear modulus decreases as the shaking
progresses  and  the  pore  fluid  pressure  increases.  It is important to note
that not all soil layers 
experience  liquefaction, as $r_u$ decreases with the depth increase.
The   excess pore fluid pressure dissipation at the lower layers is quicker than
that  of  the  upper layers. The upper layers have to dissipate their own excess
pore
pressure, however they  also receive  additional volume of water
from    lower  levels  (dissipation is upward). This leads to increase  in  pore
fluid pressure at top layers even after the shaking
has    stopped  (see  upper three plots in Figure~\ref{pp_pm_comp}).  The  soil 
settlement  and water drainage continue after the initial shaking is over mainly
due to the continuous pore fluid movement upward.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Comparison between numerical and centrifuge results.}
\label{n_vs_c_comp}



The  comparison  of  the  numerical  and the centrifuge results are presented in
Figures~\ref{disp_comp}  to  \ref{pp_comp}.  The  time  and  displacements are
multiplied  by  50  and  the  accelerations  are divided by 50, according to the
scaling laws provided in section \ref{Discussion of Scaling laws}.
Scaled results are validated using centrifuge  results,  as  presented  by
\cite{Dobry1993}.


\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/disp_comp.ps}
\end{center}
\caption{\label{disp_comp}{Time  histories  of  horizontal  displacements at the
locations  LVTD3,  LVDT4,  LVDT5 and LVDT6. Comparison between the numerical and
the experimental results from the centrifuge test, in prototype scale.}}
\end{sffamily}
\end{Large}
\end{figure}

\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/acc_comp.ps}
\end{center}
\caption{\label{acc_comp}{Time  histories  of  horizontal  accelerations  at the
locations  AH1,  AH3,  AH4  and  AH5.  Comparison  between the numerical and the
experimental results from the centrifuge test, in prototype scale.}}
\end{sffamily}
\end{Large}
\end{figure}

\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/PanagiotaTasiopoulou/Thesis/validation/Figures/pp_comp.ps}
\end{center}
\caption{\label{pp_comp}{Time  histories  of  excess  pore water pressure at the
locations  P1,  P2,  P3  and  P4.  Comparison  between  the  numerical  and  the
experimental results from the centrifuge test, in prototype scale.}}
\end{sffamily}
\end{Large}
\end{figure}



In general, it is shown  that the  physical  mechanism  of liquefaction is captured by the numerical
analysis, providing validation of numerical modeling.      
%
In      particular,     the   displacement   time  histories, and
especially  the  residual  values  of  displacement  (see Fig.~\ref{disp_comp}) 
indicate good agreement with the experimental results. The magnitude of 
cyclic component, however, is underestimated by the numerical results, a phenomena which
was also  observed in other studies \citep{Elgamal2002, Taiebat2007}.  
%
The  excess  pore  pressure  time histories (Fig.~\ref{pp_comp})  indicate  
good agreement between numerical and centrifuge results, except for the deepest layer.
%
Earlier rise
in  excess  pore  pressures is observed in the numerical analysis when compared  to the 
centrifuge test. 
Differences in liquefaction initiation and timing
affect the acceleration  time histories, 
as  can be observed in Fig. \ref{acc_comp}.
%
The  cut-off    of    acceleration,    indicative  of  liquefaction\footnote{and
providing    for    base    isolation   by   liquefaction,   as   described   by
\citet{Taiebat2009a}.}, 
start to occur deep in
the    soil    column    (location    AH5),  and this  contrasts the experimental
results.
%
These results are compatible with the development of excess pore pressure shown in
Figure~\ref{pp_comp}.
However,  there  is  a  good agreement between  the  numerical  and the
centrifuge  results  in terms of the acceleration time history at the surface of
the soil layer.




The  fact  that  liquefaction  seems  to  occur  deep in the soil column for the
numerical  case, may be attributed to the calibration of the constitutive model.
By  observing  the  stress  paths  in  Figs.~\ref{pq_pm_comp}  and
\ref{pe_pm_comp},  it  can  be  noted  that  there  is  a  quick decrease of the
effective  vertical  stress at all depths. This may be attributed to the fact that the current
constitutive  model is based only on changes in stress ratio (see Figure~\ref{CSR}) , whereas in reality
the  rate  of  excess  pore  pressure  generation  depends  on  the initial mean
effective  stress as well. This could be modified by adjusting the values of the
parameters  of  the  model with depth in order to capture this specific case. However, the
intention   of this study is to calibrate the constitutive model using the soil 
data obtained from laboratory (triaxial) tests, conducted in the
framework    of  a VELACS  project, and thus perform a real numerical prediction of
behavior   observed  in  physical  tests,  without using centrifuge test results
to (back/re-) calibrate (change) material parameters.
%
This  approach  contrasts  an  approach (often used) in  which material model parameters are
re-calibrated  (changed)   so that numerical results fit  the centrifuge test data,
while neglecting prior calibration from laboratory (triaxial) tests.
%
Possible improvement to the material model would be to render some parameters of the constitutive model
dependent  on  the  confining stress, for example the parameters that control the
dilatancy, such as $A_d$.
%


Comparison  of vertical displacements, shown in Figure~\ref{SETTLE_comp}, shows
again differences in magnitude.
%
\begin{figure}[!hbt]
\begin{Large}
\begin{sffamily}
\begin{center}
\includegraphics[width=10cm]{/home/jeremic/tex/works/Papers/2011/Validation_Coupled_Systems/SETTLE_comp.ps}
\end{center}
\caption{\label{SETTLE_comp}{Time  histories  of vertical displacements (settlement).
 Comparison  between  the  numerical  and  the
experimental  results  (at  two  surface  locations, middle -- LVDT1 and quarter
width from the side -- LVDT2 ) from the centrifuge test, in prototype scale.}}
\end{sffamily}
\end{Large}
\end{figure}
%
%Some of these differences were observed by other researchers as well.
%
%
These   differences  are  significant  and  have  been occasionally  reported in
technical meetings, but to our knowledge, have never been presented or published
in an archival journal.


There  are  a  number  possible  effects  that  can  be  used  to   explain the 
observed discrepancies  between  the  numerical  and  centrifuge results.



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\subsection{Comments on Validation Testing Using Centrifuge Modeling Results}
%\label{Discussion of Scaling laws}

Provided    here  are  comments  related  to  potential  issues with our
numerical  modeling  and with  centrifuge  modeling  results use for validation of
numerical modeling and simulation.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Use  of  Constant  Permeability.} Permeability of soil changes during
liquefaction   \citep{Shahiri2008}.    However,    the    effect    of  changing
permeability  is  not  modeled  in this  study. This modeling
simplification can potentially have large effects on results, as
with  an  increase  in permeability during liquefaction, water moves more freely
and thus more settlement occurs.
%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Boundary Conditions.}
%
Idealized boundary
conditions used in numerical simulation of centrifuge experiments 
simulate a 1D shear beam.
%
These boundary  conditions  mimic a laminar box around the contained soil.
%
For numerical model, these boundary conditions allow the (separate) vertical movement  of
the  soil  and  water  (while allowing horizontal shaking of soil and water) and
prevent any lateral expansion. However, the real conditions in the centrifuge model
involve  3D effects that, for example, may allow for a lateral
flow  and  consequently  quicker  dissipation  which leads to slower built up of
excess  pore  pressures. 
%
While  we  know  that  full 3D behavior  is present in a centrifuge laminar box,
limitations  in  instrumentation and pre-assumption of 1D behavior leave us with
no data (measurements) of such 3D behavior.
%
The only options thus is to numerically model a 1D shear box.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Variability  of  Material  in  Laboratory  Test and Centrifuge Model.}
Laboratory  tests  performed on same (similar) soil were used for calibration of
material  model  that  was then  used in modeling and simulation validation, using
centrifuge  tests.  It  is  not clear how similar soil used in those laboratory
tests  is  to the soil used in a centrifuge experiment. Only relative density is
used to describe both soils (same Nevada sand), however, other factors do affect
soil  behavior.  For  example,  sand  fabric  and  anisotropy,  both initial and
induced,  have certainly been different for laboratory and centrifuge tests, and
yet  information  about  those  differences  is  not  available  (not measured).
Differences   in  material behavior can significantly affect the final numerical
simulation  and  test  results, and that might be one of the reasons for some of
the discrepancies in this validation study.





%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Scaling of Dynamic and Dissipation Time.}
It is common to perform numerical modeling of centrifuge
tests  in  prototype scale. This is due to the fact that  centrifuge results are
published  in  prototype  scale,  while, unfortunately, the scaling factors that
have been used are not always reported. Scaling factors used to
present the experimental measurements in prototype scale correspond to 
Case  C  (see  Table  \ref{Scaling}),  resulting  in similarity of diffusion and
dynamic  time.  These  are  the  scaling  factors  applied to the current study.
However, the scaling relationships were not enforced in the centrifuge. This can 
lead  to  a possible concussion that since diffusion happens faster than pore
water    pressure  generation, more diffuses that  is  the  reason  for  slower 
pore fluid pressure rises and larger settlements in centrifuge.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Scaling of Failure Zones -- Shear Bands.}
Scaling  problems for failure of soils, where localization of deformation occurs,
can   bring additional challenges.  Sand (coarse grained particulate material) develop shear
zone that is usually 5-20 sand
particles  wide \citep{Alshibli2000}. That means that if model and original soil
are  the  same, i.e. Nevada sand with mean grain size of $0.15$~mm, the original
shear  zone  width will  be  between $5 \times 0.15{\rm mm} = 0.75$mm and $20 \times
0.15{\rm  mm}  =  3.00$mm wide, while the same width of shear zone will be carried to
the model
scale.  Ideally,  in model scale the shear zone would have to be scaled down $N$
times, which is not the case here.  This can have effect on
any   phenomena  modeled  in  centrifuge  where localized deformation occurs, as
for  example  is  the case of vertically propagating shear waves, which plastify
(and  fail) soil as they propagate. This lack of scaling of the plastified/shear
band  zone  can   influence results, particularly when soil in such
shear zone dilates and compresses, while pore fluid is being pumped out and in.


%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\paragraph{Uncertainty in Measurement Locations.}
The  accuracy  of location of  measurements  in the centrifuge can be
ambiguous. For example, if the location of the instrumentation is off by 1 cm in
the   centrifuge model, this error translates to half a meter location error for
a centrifuge experiment performed at $50$g level. 
%
In addition, dense
instrumentation  and connecting wires within  centrifuge  model affect behavior
of soils, but are not explicitly taken into account for validation.





























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\section{Conclusions}


Presented  here was a validation study that aimed to validate numerical modeling
and    simulation    of  fully  coupled  behavior of fully saturated sand, using
u-p-U formulation. Numerical
models of centrifuge tests were used for validation. An extensive discussion of
scaling  laws as they apply to validation of numerical modeling using centrifuge
test data was provided. 

Numerical modeling was, for the most part, successfully validated.
Observed  differences  have  been explained using scaling laws, mechanics of
coupled systems, and inconsistencies in boundary conditions (assumed versus actual).
%
It is noted that centrifuge modeling can be very useful in modeling of
coupled  system, provided that scaling laws are carefully taken into account and
that  phenomena  of  first  and  second  order  importance  are  prioritized and
separated.
%

On  of  the  main  conclusions  is that numerical modeling and simulation can be
successfully  used  to predict behavior of fully coupled solids (saturated soil)
and  thus  used to improve design (safety and economy), provided that there is a
clear trail of verification and validation for the numerical tool used.
%
Without  such  documented  verification  and  validation,  software  errors  and
modeling uncertainty that are (potentially) present in numerical modeling tools,
can render results unreliable and thus unusable for design.



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\section{Acknowledgement}

Author's   would  like to thanks Professor Bruce Kutter for many useful comments
and guidance provided during this research.

%\bibliography{refmech,Mahdi,zhao}
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\bibliographystyle{abbrvnat}
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%\include{Appendix}

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\end{document}
\bye
                                                                                                                                                             

