\documentclass{beamer}
%\documentclass[handout]{beamer}
%\usepackage{pgfpages}
%\pgfpagelayout{2 on 1}[letterpaper, border shrink=5mm]
\mode{\setbeamercolor{background canvas}{bg=black!3}}
\setbeamertemplate{navigation symbols}{}
\mode
{
% \usetheme{Marburg} % ima naslov i sadrzaj sa desne strane
% \usetheme{Hannover} % ima naslov i sadrzaj sa leve strane
% \usetheme{Singapore} % ima sadrzaj i tackice gore
% \usetheme{Antibes} % ima sadrzaj gore i kao graf ...
% \usetheme{Berkeley} % ima sadrzaj desno
% \usetheme{Berlin} % ima sadrzaj gore i tackice
% \usetheme{Goettingen} % ima sadrzxaj za desne strane
% \usetheme{Montpellier} % ima graf sadrzaj gore
% \usetheme{Warsaw}
% \usetheme{Warsaw}
\usetheme{Dresden}
\usecolortheme[RGB={20,0,128}]{structure}
% or ...
\setbeamercovered{transparent}
% \setbeamercovered{transparent}
% or whatever (possibly just delete it)
% \usecolortheme{albatross} % teget sa svetlim slovima
% \usecolortheme{beetle} % siva pozadina (vrh plav)
% \usecolortheme{seagull} % sivo
%%%%%%%
% \usecolortheme{BorisJeremic}
%%%%%%%
% \usecolortheme{rose}
% \usefonttheme[onlylarge]{structuresmallcapsserif}
% \usefonttheme{structuresmallcapsserif}
}
\usepackage[english]{babel}
\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
%%%% HYPERREF HYPERREF HYPERREF HYPERREF HYPERREF
%%%% HYPERREF HYPERREF HYPERREF HYPERREF HYPERREF
\definecolor{webgreen}{rgb}{0, 0.15, 0} % less intense green
\definecolor{webblue}{rgb}{0, 0, 0.15} % less intense blue
\definecolor{webred}{rgb}{0.15, 0, 0} % less intense red
%\usepackage[colorlinks=true,linkcolor=webblue,citecolor=webred,urlcolor=webgreen]{hyperref}
\usepackage{hyperref}
\hypersetup{
pdfmenubar=true,
pdftoolbar=true,
pdfpagemode={None}
}
\usepackage{pause}
% or whatever
%\usepackage{html}
%\usepackage{url}
\usepackage[latin1]{inputenc}
% or whatever
\newcommand{\ud}{{\rm d}}
\usepackage{times}
\usepackage[T1]{fontenc}
% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.
\title[ElasticPlastic Behavior of Geomaterials:
Modeling and Simulation Issues]
% (optional, use only with long paper titles)
{ElasticPlastic 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]{universitylogo}{/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{universitylogo} \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{}
% This is only inserted into the PDF information catalog. Can be left
% out.
% If you have a file called "universitylogofilename.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]{universitylogo}{/home/jeremic/BG/amblemi/ucdavis_logo_gold_lrg}
%\logo{\pgfuseimage{universitylogo}}
% \pgfdeclareimage[height=0.5cm]{universitylogo}{universitylogofilename}
% \logo{\pgfuseimage{universitylogo}}
% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
\AtBeginSubsection[]
{
\begin{frame}
\frametitle{Outline}
\tableofcontents[currentsection,currentsubsection]
% \tableofcontents[currentsection]
\end{frame}
}
% If you wish to uncover everything in a stepwise fashion, uncomment
% the following command:
%\beamerdefaultoverlayspecification{<+>}
\begin{document}
\begin{frame}
\titlepage
\end{frame}
\begin{frame}
\frametitle{Outline}
\tableofcontents
% You might wish to add the option [pausesections]
\end{frame}
% Structuring a talk is a difficult task and the following structure
% may not be suitable. Here are some rules that apply for this
% solution:
%  Exactly two or three sections (other than the summary).
%  At *most* three subsections per section.
%  Talk about 30s to 2min per frame. So there should be between about
% 15 and 30 frames, all told.
%  A conference audience is likely to know very little of what you
% are going to talk about. So *simplify*!
%  In a 20min talk, getting the main ideas across is hard
% enough. Leave out details, even if it means being less precise than
% you think necessary.
%  If you omit details that are vital to the proof/implementation,
% just say so once. Everybody will be happy with that.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Before We Start}
\subsection*{Before We Start}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Motivation}
\begin{itemize}
\vspace*{0.5cm}
\item Use well developed theory of elastoplasticity 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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Constitutive Level}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Small Deformation ElastoPlasticity}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Incremental ElastoPlasticity}
\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}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{KarushKuhnTucker 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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Explicit and Implicit Constitutive Integrations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Implicit}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 Supermatrix $\mathbb{C}$ 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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Finite Element Level}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Formulation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%
% % \vspace*{2.0truecm}
% \item
%
% \item
%
% \item
%
% \item
%
% \item
%
% \end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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*}
%
%
%
% \begin{itemize}
%
% % \vspace*{2.0truecm}
% \item
%
% \item
%
% \item
%
% \item
%
% \item
%
% \end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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*}
%
%
%
% \begin{itemize}
%
% % \vspace*{2.0truecm}
% \item
%
% \item
%
% \item
%
% \item
%
% \item
%
% \end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Statics and Dynamics}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 loadlevel parameter}
\item Proportional loading
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Hyperspherical 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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 HilberHughesTaylor, extension of Newmark with better damping \\
$1/3\le\alpha \le0, \;\;\;\gamma = {1}/{2}(12\alpha), \;\;\; \beta =
{1}/{4}(1\alpha)^2$
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 (upU, structural elements...)
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %\begin{frame}
% %\frametitle{Dynamics}
% %
% %
% %
% %
% %
% %\end{frame}
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %\begin{frame}
% %\frametitle{Dynamics}
% %
% %
% %
% %
% %
% %\end{frame}
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Examples}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Piles}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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: $py$ 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: $py$ 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: $py$ 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: $py$ 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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{$py$ 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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Pile Groups}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Piles Interaction at 2.0m ($py$)}
%\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}