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\usepackage[english]{babel}

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% does not look nice, try deleting the line with the fontenc.


\title[Elastic--Plastic  Behavior of Geomaterials:
Modeling and Simulation Issues]
% (optional, use only with long paper titles)
{Elastic--Plastic  Behavior of Geomaterials:
Modeling and Simulation Issues}

%\subtitle
%{{\tiny full set of slides available at:}\\
%%\verb{http://sokocalo.engr.ucdavis.edu/~jeremic/}
%}

\pgfdeclareimage[height=0.2cm]{university-logo}{/home/jeremic/BG/amblemi/ucdavis_logo_blue_sm}

%\author[Boris Jeremi{\'c}, CompGeoMech \includegraphics[width=8cm]{/home/jeremic/BG/amblemi/ucdavis_logo_gold_lrg}] % (optional, use only with lots of authors)
\author[Jeremi{\'c}] % (optional, use only with lots of authors)
{Boris~Jeremi{\'c}  \\  Zhaohui  Yang  (UA),  Zhao  Cheng
(EarthMechanics Inc.), Mahdi Taiebat (UBC) }
% - 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
\pgfuseimage{university-logo} \hspace*{-0.3truecm}] % (optional, but mostly needed)
{
%  \texttt{http://geomechanics.ucdavis.edu} \\
  Department of Civil and Environmental Engineering\\
  University of California, Davis}
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.

\date[GheoMat] % (optional, should be abbreviation of conference name)
{GheoMat \\
 {\small Masseria Salamina  \\
 Italy, June 2009} }
% - Either use conference name or its abbreviation.
% - Not really informative to the audience, more for people (including
%   yourself) who are reading the slides online

\subject{}
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% out.



% If you have a file called "university-logo-filename.xxx", where xxx
% is a graphic format that can be processed by latex or pdflatex,
% resp., then you can add a logo as follows:

%\pgfdeclareimage[height=0.2cm]{university-logo}{/home/jeremic/BG/amblemi/ucdavis_logo_gold_lrg}
%\logo{\pgfuseimage{university-logo}}

% \pgfdeclareimage[height=0.5cm]{university-logo}{university-logo-filename}
% \logo{\pgfuseimage{university-logo}}



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  \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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% - Talk about 30s to 2min per frame. So there should be between about
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%   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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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%  \frametitle{Motivation}
%
% \begin{itemize}
%
%   \item Designers needs the best available tools for performance assessment
%
%   \item Steady progress of numerical simulations over last 4 decades
%
%   \item Steady progress of computer hardware
%
%   \item Use of software and hardware tools for detailed (as needed) performance assessment
%
%   \item In other words: simulation tools are available
%
%   \item Need to convince practice (you) that these tools are useful,
%         robust, verified, validated, and ready to be used
%
%
% \end{itemize}
% \end{frame}






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\section{Before We Start}


\subsection*{Before We Start}

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\begin{frame}
\frametitle{Motivation}



\begin{itemize}


\vspace*{0.5cm}
\item   Use  well  developed  theory  of  elasto--plasticity  for  modeling  and
simulating geomatarials

\vspace*{0.5cm}
\item Issues at the {\it constitutive} and the {\it finite element} levels

\vspace*{0.5cm}
\item Verification and Validation is very important

\vspace*{0.5cm}
\item  There  is  no  limit  to  what  problems one can address (can numerically
simulate)


%\vspace*{2.5cm}
%\item How do we  use experimental simulations   to improve models




\end{itemize}




\end{frame}


% \subsection*{Fundamentals}
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\begin{frame}
\frametitle{Loading Process}


\begin{itemize}


\vspace*{0.5cm}
\item   Stages


\vspace*{0.5cm}
\item   Increments



\vspace*{0.5cm}
\item   Iterations


\end{itemize}






\end{frame}





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\section{Constitutive Level}







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\subsection{Small Deformation Elasto--Plasticity}


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\begin{frame}
\frametitle{Small Deformation}







\vspace*{-3cm}
\begin{figure}[!h]
\begin{center}
{\includegraphics[width=7.0cm]{/home/jeremic/tex/works/Proposals/1999/LDGeomechanics/Error_ps.pdf}}
\end{center}
\end{figure}
\vspace*{-3cm}


\begin{eqnarray*}
E_{ij} = \frac{1}{2} \left( u_{i,j} + u_{j,i} + u_{i,k}u_{k,j}\right)
\;\;\;\;\;
\mbox{;}
\;\;\;\;\;
\epsilon_{ij}=\frac{1}{2} ( u_{i,j} + u_{j,i})
\label{E01}
\end{eqnarray*}



\end{frame}

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\begin{frame}
\frametitle{Elasticity}


 \begin{itemize}

\vspace*{0.5cm}
\item     Hyperelasticity,     
$\sigma_{ij} = \partial  W / \partial \epsilon_{ij}$  (where $W$ is the strain energy function per unit
volume)

\vspace*{0.5cm}
   \item  Hypoelasticity,  direct  modeling  of nonlinear elastic deformation, not
 thermodynamically consistent


\vspace*{0.5cm}
 \item Linear and nonlinear elastic models





\end{itemize}



\end{frame}

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\begin{frame}
\frametitle{Incremental Elasto--Plasticity}


\begin{itemize}

\vspace*{0.5cm}
\item   Additive   decomposition   of  strain  $\Delta  \epsilon_{ij}  =  \Delta
\epsilon_{ij}^e + \Delta \epsilon_{ij}^p$
%

\vspace*{0.5cm}
\item    Elastic   relationship (generalized Hooke's law)  $\Delta   \sigma_{ij}   =   E_{ijkl}   \Delta
\epsilon_{kl}^{e}$

\vspace*{0.5cm}
\item  (non)  Associated  plastic  flow  rule  $\Delta  \epsilon_{ij}^p = \Delta
\lambda  \;  {  \partial  Q  }/  {\partial  \sigma_{ij}} = \Delta \lambda \;
m_{ij}(\sigma_{ij},q_{\ast})$
%

\vspace*{0.5cm}
\item Hardening/softening (isotropic/anisotropic) law 
$\Delta q_{\ast} = \Delta \lambda \; h_{\ast}(\tau_{ij},q_{\ast})$

\end{itemize}



\end{frame}
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\begin{frame}
\frametitle{Karush--Kuhn--Tucker Conditions}


\begin{itemize}

\vspace*{0.5cm}
\item Yield function
$F(\sigma_{ij}, q_{\ast}) \leq  0$

\vspace*{0.5cm}
\item Plastic consistency parameter
$\Delta \lambda \geq 0$

\vspace*{0.5cm}
\item loading -- unloading condition
$F \; \Delta \lambda = 0$

\end{itemize}





\end{frame}

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\begin{frame}
\frametitle{Midpoint Integration Algorithm}

\vspace*{-2cm}
\begin{figure}[!h]
\begin{center}
{\includegraphics[width=6.0cm,angle=-90]{/home/jeremic/tex/works/psfigures/midpointrule3.pdf}}
\end{center}
\end{figure}
\vspace*{-3cm}




\end{frame}

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\begin{frame}
\frametitle{Midpoint Integration Algorithm}


\begin{itemize}

\vspace*{0.5cm}
\item Rarely used (even if for $\alpha = 0.5$ it is second order accurate)


\vspace*{0.5cm}
\item Explicit algorithm ($\alpha = 0.0$)

\vspace*{0.5cm}
\item Implicit  algorithm ($\alpha = 1.0$)


\end{itemize}





\end{frame}

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\subsection{Explicit and Implicit Constitutive Integrations}

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\begin{frame}
\frametitle{Explicit Integration Algorithm}

\vspace*{-2cm}
\begin{figure}[!h]
\begin{center}
{\includegraphics[width=6.0cm]{/home/jeremic/tex/works/psfigures/midpointrule2.pdf}}
\end{center}
\end{figure}
\vspace*{-3cm}




\end{frame}

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\begin{frame}
\frametitle{Explicit Integration Algorithm}


\vspace*{-1cm}
Increments \hfill {\includegraphics[width=3.0cm]{/home/jeremic/tex/works/psfigures/midpointrule2.pdf}}
%\vspace*{-1cm}

\begin{equation*}
\Delta \sigma_{mn} =
E_{mnpq} \; \Delta  \epsilon_{pq} -
E_{mnpq} \;  \frac{ {}^{n}\!n_{rs} \; E_{rstu} \; \Delta \epsilon_{tu}}
{ {}^{n} \! n_{ab}\; E_{abcd} \; {}^{n}\!m_{cd}
-
 \xi_{A} h_{A} } \;
{}^{n}\!m_{pq}
\label{FEpredictor10}
\end{equation*}
%
\begin{eqnarray*}
\Delta  q_{A} = 
\left(
\frac{ {}^{n}\!n_{mn} \; E_{mnpq} \; \Delta \epsilon_{pq}}
{ \; {}^{cros} \! n_{mn}\; E_{mnpq} \; {}^{cros}\!m_{pq}
 -  \xi_{A} h_{A} }
\right)
h_{A}
\label{FEpredictor10a}
\end{eqnarray*}
%




\end{frame}

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\begin{frame}
\frametitle{Explicit Integration Algorithm}


\vspace*{-1cm}
Tangent stiffness \hfill {\includegraphics[width=3.0cm]{/home/jeremic/tex/works/psfigures/midpointrule2.pdf}}
%\vspace*{-1cm}

\begin{eqnarray*}
{}^{cont}\!E_{pqmn}^{ep} =
E_{pqmn} -
\frac{  E_{pqkl} {}^{n}\!m_{kl} {}^{n}\!n_{ij} E_{ijmn}  }
{
{}^{n}\!n_{ot} E_{otrs}\; {}^{n} \!m_{rs}
 - {}^{n} \! \xi_{A} \; h_{A}
}
\label{Continuum}
\end{eqnarray*}
%


\end{frame}

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\begin{frame}
\frametitle{Explicit Integration Algorithm}


%\vspace*{-1cm}
\hfill {\includegraphics[width=3.0cm]{/home/jeremic/tex/works/psfigures/midpointrule2.pdf}}
\vspace*{-3cm}

\begin{itemize}

\vspace*{0.5cm}
\item Relatively simple (first derivatives)


\vspace*{0.5cm}
\item Fast (single step)


\vspace*{0.5cm}
\item Inaccurate (accumulates error) 

\vspace*{0.5cm}
\item Popular (most/all commercial codes) 


\vspace*{0.5cm}
\item Works well with global explicit algorithm



\end{itemize}





\end{frame}

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%\subsection{Implicit}


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\begin{frame}
\frametitle{Implicit Integration Algorithm}

\vspace*{-2cm}
\begin{figure}[!h]
\begin{center}
{\includegraphics[width=6.0cm]{/home/jeremic/tex/works/psfigures/midpointrule2.pdf}}
\end{center}
\end{figure}
\vspace*{-3cm}




\end{frame}

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\begin{frame}
\frametitle{Implicit Integration Algorithm}




\begin{itemize}

\vspace*{0.5cm}
\item Also based on elastic predictor -- plastic corrector
${}^{n+1} \! \sigma_{ij}
=
{}^{pred} \! \sigma_{ij} -
\Delta \lambda \; E_{ijkl} \; {}^{n+1} \!m_{kl}
$

\vspace*{0.5cm}
\item Tensor of residuals used in iterations
$r_{ij} =  \sigma_{ij} - \left( {}^{pred} \! \sigma_{ij}  -
    \Delta \lambda \; E_{ijkl} \; m_{kl} \right)$

\vspace*{-1cm}
\hfill {\includegraphics[width=3.0cm]{/home/jeremic/tex/works/psfigures/midpointrule2.pdf}}
\vspace*{-2cm}


\vspace*{0.5cm}
\item Iterative increments \\
$\ud (\Delta \lambda) = 
\left( {}^{old} \! f  - 
{\bf n}^T \; \mathbb{C} \; {}^{old} {\bf r} \right)
/
\left(
{ {\bf n}^T \; \mathbb{C} \; {\bf M} } \right)$ 
%%%
\\
%%
$\left \{ \begin{array}{c}
\ud \sigma_{mn} \\
\ud q_B
\end{array} \right \}
 = 
 - \mathbb{C} 
\left(
{}^{old} {\bf r} + \ud (\Delta \lambda) {\bf m}
\right) 
$
\\ with
$\bf{n} = 
\left\{ 
\begin{array}{c}
n_{mn} \\
{\xi}_B
\end{array}
\right\} 
$, 
$
\bf{m} = 
\left\{ 
\begin{array}{c}
E_{ijkl}m_{kl} \\
-h_A
\end{array}
\right\} 
$,
%%
$
{}^{old}\bf{r} = 
\left\{ 
\begin{array}{c}
{}^{old}\sigma_{ij} \\
{}^{old}r_A
\end{array}
\right\} 
$

%%%








\end{itemize}





\end{frame}

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\begin{frame}
\frametitle{Implicit Integration Algorithm}



%\vspace*{-1cm}
\hfill {\includegraphics[width=3.0cm]{/home/jeremic/tex/works/psfigures/midpointrule2.pdf}}
\vspace*{-3cm}

\begin{itemize}

\vspace*{0.5cm}
\item    Super-matrix   $\mathbb{C}$ has different formats \\ 
         depending on a number and \\
         type of internal variables \\
%%
$
\mathbb{C} = 
\left[ \begin{array}{c}
I^s_{ijmn} + \Delta \lambda E_{ijkl}\frac{ \partial m_{kl} }{\partial \sigma_{mn}} 
\end{array} \right]^{-1}
$
%%
%%
$
\mathbb{C} = 
\left[ \begin{array}{cc}
I^s_{ijmn} + \Delta \lambda E_{ijkl}\frac{ \partial m_{kl} }{\partial \sigma_{mn}} 
&  \Delta \lambda E_{ijkl} \frac{ \partial m_{kl} }{\partial q} \\
- \Delta \lambda \frac{ \partial h }{\partial \sigma_{ij}} 
& 1 - \Delta \lambda \frac{ \partial h }{\partial q}
\end{array} \right]^{-1}
$
%% \\ 
%%  %%
%%  $
%%  \mathbb{C} = 
%%  \left[ \begin{array}{cc}
%%  I^s_{ijmn} + \Delta \lambda E_{ijkl}\frac{ \partial m_{kl} }{\partial \sigma_{mn}} 
%%  &  \Delta \lambda E_{ijkl} \frac{ \partial m_{kl} }{\partial \alpha_{mn}} \\
%%  - \Delta \lambda \frac{ \partial h_{mn} }{\partial \sigma_{ij}} 
%%  & I^s_{ijmn} - \Delta \lambda \frac{ \partial h_{mn} }{\partial \alpha_{ij}}
%%  \end{array} \right]^{-1}
%%  $
%%  // 
%%
%%  $
%%  \mathbb{C} = 
%%  \left[ \begin{array}{ccc}
%%  I^s_{ijmn} + \Delta \lambda E_{ijkl}\frac{ \partial m_{kl} }{\partial \sigma_{mn}}
%%  &  \Delta \lambda E_{ijkl} \frac{ \partial m_{kl} }{\partial q}
%%  &  \Delta \lambda E_{ijkl} \frac{ \partial m_{kl} }{\partial \alpha_{mn}} \\
%%  - \Delta \lambda \frac{ \partial h }{\partial \sigma_{ij}} 
%%  &  1 - \Delta \lambda \frac{ \partial h }{\partial q} 
%%  &  - \Delta \lambda \frac{ \partial h }{\partial \alpha_{ij}} \\
%%  - \Delta \lambda \frac{ \partial h_{mn} }{\partial \sigma_{ij}} 
%%  &  - \Delta \lambda \frac{ \partial h_{mn} }{\partial q} 
%%  &  I^s_{ijmn} - \Delta \lambda \frac{ \partial h_{mn} }{\partial \alpha_{ij}}
%%  \end{array} \right]^{-1}
%%  $
%%  // 
\\
%%
$
\mathbb{C} = 
\left[ \begin{array}{ccc}
I^s_{ijmn} + \Delta \lambda E_{ijkl}\frac{ \partial m_{kl} }{\partial \sigma_{mn}}
&  \Delta \lambda E_{ijkl} \frac{ \partial m_{kl} }{\partial z_{mn}}
&  \Delta \lambda E_{ijkl} \frac{ \partial m_{kl} }{\partial \alpha_{mn}} \\
- \Delta \lambda \frac{ \partial h^z }{\partial \sigma_{ij}} 
&  I^s_{ijmn} - \Delta \lambda \frac{ \partial h^z }{\partial z_{mn}} 
&  - \Delta \lambda \frac{ \partial h^z }{\partial \alpha_{ij}} \\
- \Delta \lambda \frac{ \partial h^{\alpha}_{mn} }{\partial \sigma_{ij}} 
& - \Delta \lambda \frac{ \partial h^{\alpha}_{mn} }{\partial z_{mn}} 
&  I^s_{ijmn} - \Delta \lambda \frac{ \partial h^{\alpha}_{mn} }{\partial \alpha_{ij}}
\end{array} \right]^{-1}
$
%%




\end{itemize}





\end{frame}

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\begin{frame}
\frametitle{Implicit Integration Algorithm}




\vspace*{-1cm}
Consistent (algorithmic) stiffness \hfill {\includegraphics[width=3.0cm]{/home/jeremic/tex/works/psfigures/midpointrule2.pdf}}
%\vspace*{-1cm}

\begin{eqnarray*}
\left \{ \begin{array}{c}
\ud \sigma_{ij} \\
\ud q_A
\end{array} \right \}
 =
\left \{
\mathbb{C}
-
\frac{ \mathbb{C} {\bf m} {\bf n}^T \mathbb{C} }
{ {\bf n}^T \; \mathbb{C} \; {\bf m} } 
\right \}
\left \{ \begin{array}{c}
E_{ijmn} \; \ud \epsilon^{pred}_{mn} \\
0
\end{array} \right \}
\end{eqnarray*}
%




\end{frame}

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\begin{frame}
\frametitle{Implicit Integration Algorithm}


%\vspace*{-1cm}
\hfill {\includegraphics[width=3.0cm]{/home/jeremic/tex/works/psfigures/midpointrule2.pdf}}
\vspace*{-3cm}

\begin{itemize}

\vspace*{0.2cm}
\item Relatively complicated (first and \\
      second derivatives, inverse)


\vspace*{0.2cm}
\item Relatively slow (but improves global \\
       Newton iterations)


\vspace*{0.2cm}
\item Accurate (consistency condition satisfied at the end, within tolerance)

\vspace*{0.2cm}
\item Popular for research 

\vspace*{0.2cm}
\item Unpopular in commercial codes (except simple material models) 


\vspace*{0.2cm}
\item Designed to work with global Newton algorithm



\end{itemize}





\end{frame}

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\section{Finite Element Level}








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\subsection{Formulation}


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\begin{frame}
\frametitle{Principle of Virtual Displacements}


%
%
\begin{equation*}
\int_{V } \sigma_{ij} \;
\delta \epsilon_{ij} \;  dV
=
\int_{V } 
\left(
f^{B}_{i} - \rho \ddot{u}_{i}
\right)
 \;
\delta u_{i} \; dV
+
\int_{S } f^{S}_{i} \;
\delta u_{i} \; dS
\end{equation*}
%
%  %


%  
%  
%  \begin{itemize}
%  
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\end{frame}

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\begin{frame}
\frametitle{Discretization}


%
\begin{equation*}
u \approx \hat{u}_{a} = H_{I} \bar{u}_{Ia}
\label{FED02}
\end{equation*}
%

%
\begin{eqnarray*}
\epsilon_{ab} \; \approx \; \hat{e}_{ab}
&=&
\frac{1}{2}\left(\hat{u}_{a,b}+\hat{u}_{b,a}\right)
=  \nonumber \\ &=&
\frac{1}{2}  \left(
\left(H_{I} \; \bar{u}_{Ia}\right)_{,b}
+
\left(H_{I} \; \bar{u}_{Ib}\right)_{,a}
\right)
=  \nonumber \\ &=&
\frac{1}{2}  \left(
\left( H_{I,b} \; \bar{u}_{Ia} \right)
+
\left( H_{I,a} \; \bar{u}_{Ib} \right)
\right)
\label{FED04}
\end{eqnarray*}
%

%  
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\end{frame}

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\begin{frame}
\frametitle{FEM Equations}


%
\begin{eqnarray*}
\bigcup_{(m)} {}^{(m)}\!  M_{IacJ} \; \ddot{\bar{u}}_{Jc}
+
\bigcup_{(m)} {}^{(m)}\! K_{IacJ}   \; \bar{u}_{Jc}
%
= 
%
\bigcup_{m} {}^{(m)}\! F_{Ia}^{B}
%
+
%
\bigcup_{m} {}^{(m)}\! F_{Ia}^{S}
%
\end{eqnarray*}


 %
\begin{eqnarray*}
{}^{(m)}\! M_{IacJ}
=
\int_{V^{m} } H_{J}  \; \delta_{ac} \; \rho  \;  H_{I} \;  dV^{m} 
\;\;\;\;
\mbox{;}
\;\;\;\;
{}^{(m)}\! F_{Ia}^{B}
=
\int_{V^{m} } f^{B}_{a} \; H_{I} \; dV^{m}
\end{eqnarray*}

%
%
\begin{eqnarray*}
{}^{(m)}\! K_{IacJ}
=
\int_{V^{m} }  H_{I,b} \; E_{abcd} \; H_{J,d} \;  dV^{m}
\;\;\;\;
\mbox{;}
\;\;\;\;
{}^{(m)}\! F_{Ia}^{S}
=
\int_{S^{m} } f^{S}_{a} \; H_{I} \; dS^{m}
\end{eqnarray*}



%

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\end{frame}

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\subsection{Statics and Dynamics}


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\begin{frame}
\frametitle{Residual Force Equation in Statics}


%
\begin{equation*}
{r_i}({u_j},\lambda) =
{f}^{int}_i({u_j}) - \lambda {f}^{ext}_{i} = 0
\end{equation*}
%



\begin{itemize}

%   \vspace*{2.0truecm}
   \item ${f}^{int}_i({u_j})$ are the internal forces which are functions of the
   displacements ${u_j}$,

   \item ${f}^{ext}_{i}$ is a {\em fixed external  loading  vector}  

   \item   $\lambda$  is  a  {\em load--level parameter}

   \item  Proportional  loading 


\end{itemize}



\end{frame}

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\begin{frame}
\frametitle{Advancing the Solution}

\vspace*{-1cm}
\begin{figure}[!h]
\begin{center}
{\includegraphics[width=7.0cm]{/home/jeremic/tex/works/psfigures/ArcLengthMethod01.pdf}}
\end{center}
\end{figure}
\vspace*{-2cm}

 


\end{frame}

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\begin{frame}
\frametitle{Hyper--spherical  Constraint}

      
\begin{equation*}
s =  \int ds
\;\;\;\;\;
\mbox{where}
\;\;\;\;\;
ds  =
\sqrt{ \frac{\psi_{u}^{2}}{u_{ref}^{2}} d{u_i} {S_{ij}} d{ u_j}
+
d \lambda^{2} \psi_{f}^{2} }
\end{equation*}

or, in incremental form:

\begin{equation*}
a =
(\Delta s)^2 - (\Delta l)^2 =
\left( \frac{\psi_{u}^{2}}{u_{ref}^{2}} 
\Delta {u_i}
{S_{ij}} 
\Delta {u_i}
+
\Delta \lambda^{2} \psi_{f}^{2} \right)
-
\left(\Delta l \right)^2
\end{equation*}





\end{frame}

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\begin{frame}
\frametitle{Specializations}


\vspace*{-1cm}
\begin{figure}[!h]
\begin{center}
{\includegraphics[width=4.0cm]{/home/jeremic/tex/works/psfigures/ArcLengthMethod02.pdf}}
{\includegraphics[width=4.0cm]{/home/jeremic/tex/works/psfigures/ArcLengthMethod03.pdf}}
\end{center}
\end{figure}

\vspace*{-1.5cm}


\begin{itemize}

\item Coefficients  $\psi_{u}$  and $\psi_{f}$ may not be simultaneously zero

\item      If  ${S_{ij}}  = {I_{ij}}$ and $u_{ref} = 1$ $\rightarrow$ {arclength
method}

\item  If  ${S_{ij}}  =  {K}^{t}_{ij}$   and  $\psi_{f}  \equiv 0$ $\rightarrow$
{external work constraint}

\item  If   $ \psi_{u} \equiv 0$ and $\psi_{f}   \equiv   1$  $\rightarrow$ load
control

\item      If      $ \psi_{u} \equiv 1$, $\psi_{f}   \equiv   0$ and ${S_{ij}} =
{I_{ij}}$ $\rightarrow$ generalized displacement control

\end{itemize}



\end{frame}

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\begin{frame}
\frametitle{Following the Equilibrium Path in Statics}

      

\begin{itemize}

\item Family of Newton methods (full, initial stress, modified...)

\item  Traversing   equilibrium  path  in positive sense (positive external work
criterion; angle criterion)


\item Accuracy control


\item  Numerical stability


\item Automatic increments


\item Convergence criteria (absolute, relative, force and/or displacement and/or
energy based)



\end{itemize}




\end{frame}

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\begin{frame}
\frametitle{Transient Integration Algorithms}

\begin{itemize}

\vspace*{0.3cm}
\item Finite differences, simple, but inacurate

\vspace*{0.3cm}
\item Wilson $\theta = 1.37$, too much numerical damping

\vspace*{0.3cm}
\item     Newmark, controllable numerical damping, period elongation    \\ 
$\gamma     \ge     {1}/{2},     \;\;\;    \beta    = {1}/{4}(\gamma+{1}/{2})^2$

\vspace*{0.3cm}
\item   Hilber--Hughes--Taylor, extension of Newmark with better damping  \\
$-1/3\le\alpha   \le0,    \;\;\;\gamma  =  {1}/{2}(1-2\alpha),  \;\;\;  \beta  =
{1}/{4}(1-\alpha)^2$



\end{itemize}


\end{frame}

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\begin{frame}
\frametitle{Dynamic Analysis}



\begin{itemize}

\vspace*{0.3cm}
 \item   Stability   (artificial   introduction   of   higher   frequencies   by
 discretization process)

\vspace*{0.3cm}
  \item Accuracy, conservation of energy and period

\vspace*{0.3cm}
 \item  Time  step  choice (the shorter the better, unless too many (artificial)
 high frequencies are present).

\vspace*{0.3cm}
 \item multiple DOF type systems (u-p-U, structural elements...)


\end{itemize}

\end{frame}

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% %\begin{frame}
% %\frametitle{Dynamics}
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% %\begin{frame}
% %\frametitle{Dynamics}
% %
% %
% %
% %
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\section{Examples}

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\subsection{Piles}

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\begin{frame}
 \frametitle{Single Pile in Layered Soils: Model}


\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/SinglePile_CS_color.pdf}
\hfill
\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/spUFmesh.pdf}
\end{center}
%\vspace*{-0.3cm}
\end{figure}

\end{frame}


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\begin{frame}
 \frametitle{Available Data for Prototype}


 \begin{itemize}
   \item Sand:
    \begin{itemize}
     \item friction angle $\phi$ of  $37.1^{o}$,
     \item Shear  modulus  at  a depth of 13.7 m of 8960 kPa ($E_o$ = 17400 kPa),
     \item Poisson ratio  of  0.35   
     \item Unit  weight  of  14.50  kN/$m^3$. 
     \item Dilation angle   $0^{o}$
    \end{itemize}
   \item Clay (made up)
     \begin{itemize}
       \item Shear strength $21.7$~kPa
       \item Young's modulus $11000$~kPa
       \item Poisson ratio $0.45$
       \item Unit weight $13.7~{\rm kN/m^3}$
     \end{itemize}

 \end{itemize}


\end{frame}
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\begin{frame}
 \frametitle{Single Pile in Sand: M, Q, p}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=7.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/MQP_dsand_UFmid2hdp0.jpg}
\end{center}
\end{figure}
\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Single Pile in Clay: M, Q, p}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=7.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/MQP_sfclay_UF2hdp0.jpg}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
 \frametitle{Single Pile in Sand with Clay Layer: M, Q, p}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=7.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/MQP_sand_wsfclay_UFmid2hdp0.jpg}
\end{center}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
 \frametitle{Single Pile in Clay with Sand Layer: M, Q, p}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=7.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/MQP_sfclay_sand_UF2hdp0.jpg}
\end{center}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Single Pile in Sand: $p-y$ Response}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=6.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/p_y_dsand_UFmid2hdp0.jpg}
\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/MQP_dsand_UFmid2hdp0.jpg}
\hspace*{-1cm}
\end{center}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Single Pile in Clay: $p-y$ Response}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=6.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/p_y_sfclay_UF2hdp0.jpg}
\includegraphics[width=3.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/MQP_sfclay_UF2hdp0.jpg}
\hspace*{-1cm}
\end{center}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Single Pile in Sand with Clay Layer: $p-y$ Response}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=6.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/p_y_sand_sfclay_UFmid2hdp0.jpg}
\includegraphics[width=3.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/MQP_sand_wsfclay_UFmid2hdp0.jpg}
\hspace*{-1cm}
\end{center}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Single Pile in Clay with Sand Layer: $p-y$ Response}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=6.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/p_y_sfclay_sand_UF2hdp0.jpg}
\includegraphics[width=3.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/MQP_sfclay_sand_UF2hdp0.jpg}
\hspace*{-1cm}
\end{center}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
 \frametitle{$p-y$ Pressure Ratio Reduction for Layered Soils}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=6.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/P_ReductionRatio_sand3casesdp0_065.jpg}
\includegraphics[width=3.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/MQP_sand_wsfclay_UFmid2hdp0.jpg}
\hspace*{-1cm}
\end{center}
\end{figure}
\end{frame}
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\subsection{Pile Groups}


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\begin{frame}
 \frametitle{Pile Group Simulations}

\begin{figure}[!hbpt]
\vspace*{-1.0cm}
\begin{center}
\hspace*{-1.0cm}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/PileGroups/p4X3_Joey_iso.pdf}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/PileGroups/Plastified_3X3pg_skewCOLOR.pdf}
\end{center}
\end{figure}

\end{frame}

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\begin{frame}
 \frametitle{Bending Moments}

\begin{figure}[!hbpt]
%\vspace*{-0.3cm}
\begin{center}
\includegraphics[width=7.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/PileGroups/4x3PileGroupBendingMoments.jpg}
\end{center}
\end{figure}
%\vspace*{-1.0cm}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
 \frametitle{Out of Plane Effects}

\begin{figure}[!hbpt]
\vspace*{-0.3cm}
\begin{center}
\includegraphics[width=4.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/PileGroups/Mx_4x3_4piles_com_dense.jpg}
\includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/PileGroups/p4X3_deformedPile_Y.jpg}
\end{center}
\end{figure}
\vspace*{-1.0cm}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
 \frametitle{Load Distribution per Pile}

\begin{figure}[!h]
\vspace*{-6.3cm}
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/PileGroups/LoadRatio_per_pile_Dense_pg4x3.pdf}
\end{center}
\end{figure}
\vspace*{-8.3cm}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
 \frametitle{Piles Interaction at -2.0m ($p-y$)}




%\vspace*{-1.3cm}
%\vspace*{-1.3cm}
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=10cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/PileGroups/pg4x3_p_y_allpiles_20.jpg}
\end{center}
\end{figure}

%\begin{large}
%\begin{itemize}
%\item  Note the difference in response curves (cannot scale single pile response
%for multiple piles)
%\end{itemize}
%\end{large}


\end{frame}



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\section{Summary}



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


\begin{itemize}


\vspace*{0.5cm}
     \item Importance of consistent formulation, material modeling and implementation

\vspace*{0.5cm}
     \item Verified, validate models and simulations tools used for prediction of behavior

\vspace*{0.5cm}
     \item Program and examples available in public domain (Author's web site)

%
%\item Models available (some now, some later) 

%\vspace*{0.5cm}
\end{itemize}

\end{frame}
%--

\end{document}


