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\title{Stochastic ElasticPlastic Finite Element Method for
Performance Risk Simulations}
%\subtitle
%{Include Only If Paper Has a Subtitle}
%\author[Author, Another] % (optional, use only with lots of authors)
%{F.~Author\inst{1} \and S.~Another\inst{2}}
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% affiliation.
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\pgfdeclareimage[height=0.35cm]{UAlogo}{/home/jeremic/BG/amblemi/uakron_logo_03}
\author[Jeremi{\'c} and Sett] % (optional, use only with lots of authors)
{Boris~Jeremi{\'c}\inst{1} \and Kallol Sett\inst{2}}
%  Give the names in the same order as the appear in the paper.
%  Use the \inst{?} command only if the authors have different
% affiliation.
\institute[Computational Geomechanics Group \hspace*{0.3truecm}
\pgfuseimage{universitylogo}\hspace*{0.1truecm}\pgfuseimage{UAlogo} \hspace*{0.3truecm}] % (optional, but mostly needed)
{ \inst{1}University of California, Davis
\and
\inst{2}University of Akron, Ohio}
%  Use the \inst command only if there are several affiliations.
%  Keep it simple, no one is interested in your street address.
\date[ICASP11] % (optional, should be abbreviation of conference name)
{ICASP \\
{\small Z{\"u}rich, Switzerland \\
August 2011} }
\subject{}
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\frametitle{Outline}
\tableofcontents
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% 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).
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%  Talk about 30s to 2min per frame. So there should be between about
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% 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.
\section{Motivation}
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%\subsection{Risk and Civil Engineering}
\subsection{}
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\begin{frame}
\frametitle{Determining Risk for Civil Engineering Object Behavior}
\begin{itemize}
%\vspace*{0.4cm}
\item Risk: inherent, intrinsic, constitutive part of civil engineering
\vspace*{0.4cm}
\item Uncertain loads (!)
\vspace*{0.4cm}
\item Uncertain materials (!!)
\vspace*{0.4cm}
\item Uncertain human factor (!)
\end{itemize}
\begin{equation}
{\bf M} \ddot{\bf u}
+
{\bf C} \dot{\bf u}
+
{\bf K} {\bf u}
=
{\bf F}
\nonumber
\end{equation}
\end{frame}
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% \begin{frame}
%
% \frametitle{Types of Uncertainties}
%
%
%
% \begin{itemize}
%
%
%
% \item Epistemic uncertainty  due to lack of knowledge
%
% \begin{itemize}
%
% \item Can be reduced by collecting more data
%
% \item Mathematical tools are not well developed
% \item tradeoff with aleatory uncertainty
%
% \end{itemize}
%
%
% \vspace*{0.2cm}
% \item Aleatory uncertainty  inherent variation of physical system
%
% \begin{itemize}
%
% \item Can not be reduced
%
% \item Has highly developed mathematical tools
%
% \end{itemize}
%
%
%
% \end{itemize}
%
%
%
%
% \vspace*{0.5cm}
% \begin{figure}[!hbpt]
% %\nonumber
% \begin{center}
% %\begin{center}
% %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
% \includegraphics[height=4cm,angle=90]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/uncertain03.pdf}
% %
% %\mbox{\tiny{Lambe, T. W. and Whitman, R. V.,1969. Soil Mechanics. New York, John Wiley \& Sons}}
% %
% \end{center}
% %\begin{flushright}
% %Soil Variability in Relatively \\ Homogeneous Soil Deposit \\ (Clay Deposit of the Valley \\ of Mexico at a Typical \\ Spot in Mexico City)
% %\includegraphics[width=14cm]{TypicalSoilCOV.jpg}
% %
% %\nonumber
% %\caption{DruckerPrager Associative Linear Hardening Response with Random $G$}
% %{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
% %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
% %\end{flushright}
% %\end{center}
% \end{figure}
%
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% \begin{frame}
%
% \frametitle{Ergodicity}
%
%
%
% \begin{itemize}
%
%
%
% \item Exchange ensemble averages for time averages
%
% \vspace*{0.2cm}
% \item Is soil elastoplasticity ergodic?
%
% \begin{itemize}
%
% \item Can soil elasticplastic statistical properties be obtained by
% temporal averaging?
%
%
% \item Will soil elasticplastic statistical properties "renew" at each
% occurrence?
%
% \item Are soil elasticplastic statistical properties statistically
% independent?
%
% \end{itemize}
%
%
% \item Claim in literature that structural nonlinear behavior is nonergodic
% while earthquake characteristics are (?!)
%
% \item However, earthquake characteristics is representing mechanics (fault slip)
% on a different scale...
%
% \end{itemize}
%
%
% \end{frame}
%
%
%
%
%
%
%
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\begin{frame}
\frametitle{Material Behavior Inherently Uncertain}
%\begin{itemize}
%\vspace*{0.5cm}
%\item
%Material behavior is inherently uncertain (concrete, metals, soil, rock,
%bone, foam, powder etc.)
\begin{itemize}
\vspace*{0.5cm}
\item Spatial \\
variability
\vspace*{0.5cm}
\item Pointwise \\
uncertainty \\
\begin{itemize}
\item testing \\
error \\
\item transformation \\
error
\end{itemize}
\end{itemize}
% \vspace*{0.5cm}
% \item Failure mechanisms related to spatial variability (strain localization and
% bifurcation of response)
%
% \vspace*{0.5cm}
% \item Inverse problems
%
% \begin{itemize}
%
% \item New material design, ({\it pointwise})
%
% \item Solid and/or structure design (or retrofits), ({\it spatial})
%
% \end{itemize}
%\end{itemize}
\vspace*{5cm}
\begin{figure}[!hbpt]
%\nonumber
%\begin{center}
\begin{flushright}
%\includegraphics[height=5.0cm]{/home/jeremic/tex/works/Conferences/2006/KragujevacSEECCM06/Presentation/MGMuzorak01.jpg}
\includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/FrictionAngleProfile.jpg}
\\
\mbox{(Mayne et al. (2000) }
\end{flushright}
%\end{center}
%\end{center}
\end{figure}
\end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame} \frametitle{\Large{Motivation}}
%
%
% \center{Typical Coefficients of Variation of Different Soil Properties}
%
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/TableTypicalCOV.jpg}
% \end{center}
% \end{figure}
% \flushright{(After Lacasse and Nadim 1996)}
%
%
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%
% \end{frame}
%
%
%
%
%
%
%
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\section{Probabilistic ElastoPlasticity}
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\subsection{PEP Formulations}
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\begin{frame}
\frametitle{Uncertainty Propagation through Constitutive Eq.}
%
\begin{itemize}
\item Incremental elpl constitutive equation
$\displaystyle \frac{d\sigma_{ij}}{dt} = D_{ijkl} \displaystyle \frac{d\epsilon_{kl}}{dt}$
%\begin{normalsize}
%
% \begin{equation}
% \nonumber
% \frac{d\sigma_{ij}}{dt} = D_{ijkl} \frac{d\epsilon_{kl}}{dt}
% \end{equation}
\begin{eqnarray}
\nonumber
D_{ijkl} = \left\{\begin{array}{ll}
%
D^{el}_{ijkl}
%
%
\;\;\; & \mbox{\large{~for elastic}} \\
%
\\
%
D^{el}_{ijkl}

\frac{\displaystyle D^{el}_{ijmn} m_{mn} n_{pq} D^{el}_{pqkl}}
{\displaystyle n_{rs} D^{el}_{rstu} m_{tu}  \xi_* r_*}
\;\;\; & \mbox{\large{~for elasticplastic}}
%
\end{array} \right.
\end{eqnarray}
%\end{normalsize}
%\vspace{0.5cm}
% \item Nonlinear coupling in the ElPl modulus
\item What if all (any) material parameters are uncertain
\item Since material {\bf is} inherently spatially variable and uncertain at the
point, PEP and SEPFEM methods were developed
% \item Focus on 1D $\rightarrow$ a nonlinear ODE with random coefficient and random forcing
%
%
%
% \begin{eqnarray}
% \nonumber
% \frac{d\sigma(x,t)}{dt} &=& \beta(\sigma(x,t),D^{el}(x),q(x),r(x);x,t) \frac{d\epsilon(x,t)}{dt} \\
% \nonumber
% &=& \eta(\sigma,D^{el},q,r,\epsilon; x,t) \mbox{\ \ \ \ with an I.C. $\sigma(0)=\sigma_0$}
% \end{eqnarray}
%
\end{itemize}
%
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\end{frame}
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% \begin{frame}
%
% \frametitle{Problem Statement}
%
%
%
%
%
% \begin{itemize}
%
% \item Incremental 3D elasticplastic stressstrain:
% %
% %
%
% \begin{equation}
% \nonumber
% \frac{ d\sigma_{ij}}{d t} = \left \{
% D^{el}_{ijkl}
% 
% \frac{\displaystyle D^{el}_{ijmn} m_{mn} n_{pq} D^{el}_{pqkl}}
% {\displaystyle n_{rs} D^{el}_{rstu} m_{tu}  \xi_* r_*}
% \right \}
% \frac{ d\epsilon_{kl}}{d t}
% \end{equation}
%
%
%
%
%
% \item Focus on 1D (3D is also available) $\rightarrow$ a nonlinear ODE with random coefficient
% (material) and random forcing ($\epsilon$)
% %
% %
% %
% \begin{eqnarray}
% \nonumber
% \frac{d\sigma(x,t)}{dt} &=& \beta(\sigma(x,t),D^{el}(x),q(x),r(x);x,t) \frac{d\epsilon(x,t)}{dt} \\
% \nonumber
% &=& \eta(\sigma,D^{el},q,r,\epsilon; x,t)
% \end{eqnarray}
% %
% with initial condition $\sigma(0)=\sigma_0$
%
% \end{itemize}
%
% \end{frame}
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{Solution to Probabilistic ElasticPlastic Problem}
\begin{itemize}
\item Use of stochastic continuity (Liouiville) equation (Kubo 1963)
%\vspace{0.1cm}
\item With cumulant expansion method (Kavvas and Karakas 1996)
%\vspace{0.1cm}
\item To obtain ensemble average form of Liouville Equation
%\vspace{0.1cm}
\item Which, with van Kampen's Lemma (van Kampen 1976): ensemble average of
phase density is the probability density
%\vspace{0.1cm}
\item Yields EulerianLagrangian form of the Forward Kolmogorov
(FokkerPlanckKolmogorov) equation
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{EulerianLagrangian FPK Equation}
%
%\begin{itemize}
% 3D
\begin{footnotesize}
\begin{eqnarray}
\nonumber
\lefteqn{\displaystyle \frac{\partial P(\sigma_{ij}(x_t,t), t)}{\partial t} = \displaystyle \frac{\partial}{\partial \sigma_{mn}}
\left[ \left\{\left< \vphantom{\int_{0}^{t}} \eta_{mn}(\sigma_{mn}(x_t,t), D_{mnrs}(x_t), \epsilon_{rs}(x_t,t))\right> \right. \right.} \\
\nonumber
&+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[\displaystyle \frac{\partial \eta_{mn}(\sigma_{mn}(x_t,t), D_{mnrs}(x_t),
\epsilon_{rs}(x_t,t))} {\partial \sigma_{ab}}; \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta_{ab} (\sigma_{ab}(x_{t\tau}, t\tau), D_{abcd}(x_{t\tau}), \epsilon_{cd}(x_{t\tau}, t\tau)
\vphantom{\int_{0}^{t}} \right] \right \} P(\sigma_{ij}(x_t,t),t) \right] \\
\nonumber
&+& \displaystyle \frac{\partial^2}{\partial \sigma_{mn} \partial \sigma_{ab}} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[
\vphantom{\int_{0}^{t}} \eta_{mn}(\sigma_{mn}(x_t,t), D_{mnrs}(x_t), \epsilon_{rs}(x_t,t)); \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta_{ab} (\sigma_{ab}(x_{t\tau}, t\tau), D_{abcd}(x_{t\tau}), \epsilon_{cd}(x_{t\tau}, t\tau))
\vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma_{ij}(x_t,t),t) \right]
\end{eqnarray}
\end{footnotesize}
% 1D % 1D
% 1D \begin{footnotesize}
% 1D \begin{eqnarray}
% 1D \nonumber
% 1D &&\displaystyle \frac{\partial P(\sigma(x_t,t), t)}{\partial t}=
% 1D  \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D^{el}(x_t),
% 1D q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\
% 1D \nonumber
% 1D &+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t),
% 1D \epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right. \\
% 1D \nonumber
% 1D & & \left. \left. \left. \eta(\sigma(x_{t\tau},t\tau), D^{el}(x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}),
% 1D \epsilon(x_{t\tau},t\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} P(\sigma(x_t,t),t) \right] \\
% 1D \nonumber
% 1D &+& \displaystyle \frac{\partial^2}{\partial \sigma^2} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}}
% 1D \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t), \epsilon(x_t,t)); \right. \right. \right. \\
% 1D \nonumber
% 1D & & \left. \left. \left. \eta (\sigma(x_{t\tau},t\tau), D^{el}(x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}),
% 1D \epsilon(x_{t\tau},t\tau)) \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma (x_t,t),t) \right] \\
% 1D \nonumber
% 1D \end{eqnarray}
% 1D
% 1D \end{footnotesize}
\end{frame}
%
% \item 6 equations
%
% \item Complete description of 3D probabilistic stressstrain response
%
% \end{itemize}
%
%
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\begin{frame}
\frametitle{EulerianLagrangian FPK Equation}
\begin{itemize}
\item Advectiondiffusion equation
%
\begin{equation}
\nonumber
\frac{\partial P(\sigma_{ij},t)}{\partial t}
=
\frac{\partial}{\partial\sigma_{ab}}
\left[N_{ab}^{(1)}P(\sigma_{ij},t)

\frac{\partial}{\partial \sigma_{cd}}
\left\{N_{abcd}^{(2)} P(\sigma_{ij},t)\right\} \right]
\end{equation}
%
\item Complete probabilistic description of response
\item Solution PDF is {\bf secondorder exact} to covariance of time (exact mean and variance)
\item Deterministic equation in probability density space
\item Linear PDE in probability density space
$\rightarrow$ simplifies the numerical solution process
\item Applicable to any elasticplasticdamage material model (only coefficients $N_{ab}^{(1)}$
and $N_{abcd}^{(2)}$ differ)
%\vspace*{0.2truecm}
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Probabilistic ElasticPlastic Response}
\begin{figure}[!hbpt]
%\nonumber
%\begin{flushleft}
\begin{center}
%\includegraphics[height=3.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFElastic_RandomGm.pdf}
%\hspace*{0.3cm}
%\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Conferences/2011/ICASP11_Zurich/Present/Probabilistic_El_Pl_example.jpg}
\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Conferences/2011/ICASP11_Zurich/Present/PDF_PlotEd.pdf}
%
%\end{flushleft}
%\begin{flushright}
%\includegraphics[height=9cm]{ReasonOfNotMatchingMonteCarlom.pdf}
%
%\nonumber
%\caption{DruckerPrager Associative Linear Hardening Response with Random $G$}
%{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%\end{flushright}
\end{center}
\end{figure}
%Probabilistic Elastic \hspace*{2cm} Probabilistic ElasticPlastic
%%$$ = 2.5 MPa;
%%Std. Deviation$[G]$ = 0.5 MPa
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\end{frame}
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%\subsection{Boundary Value Problem}
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\subsection[SEPFEM]{Stochastic ElasticPlastic Finite Element Method}
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\begin{frame}
\frametitle{Governing Equations \& Discretization Scheme}
\begin{itemize}
\item Governing equations:
%
%
\begin{equation}
\nonumber
A\sigma = \phi(t) ~~~~~~~ Bu = \epsilon ~~~~~~~ \sigma = \textcolor{mycolor}{\bf D} \epsilon
\end{equation}
\item Discretization (spatial and stochastic) schemes
\begin{itemize}
\item Input random field material properties ($\textcolor{mycolor}{\bf D}$) $\rightarrow$
KarhunenLo{\`e}ve (KL) expansion, optimal expansion, error minimizing property
\item Unknown solution random field ($u$) $\rightarrow$ Polynomial Chaos (PC)
expansion
\item Deterministic spatial differential operators ($A$ \& $B$) $\rightarrow$
Regular shape function method with Galerkin scheme
\end{itemize}
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Spectral Stochastic ElasticPlastic FEM}
\begin{itemize}
\item Minimizing norm of error of finite representation using Galerkin
technique (Ghanem and Spanos 2003):
\vspace*{0.6truecm}
\begin{flushright}
\begin{equation}
\nonumber
\sum_{n = 1}^N K_{mn} d_{ni} + \sum_{n = 1}^N \sum_{j = 0}^P d_{nj} \sum_{k = 1}^M C_{ijk} K'_{mnk} = \left< F_m \psi_i[\{\xi_r\}] \right >
\end{equation}
\end{flushright}
% \begin{itemize}
%
% \vspace*{0.5cm}
% \item Final eqn.:
%
% \vspace*{0.4cm}
% \begin{flushright}
% \begin{normalsize}
% \begin{equation}
% \nonumber
% \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sum_{n = 1}^N K_{mn} d_{ni} + \sum_{n = 1}^N \sum_{j = 0}^P d_{nj} \sum_{k = 1}^M C_{ijk} K'_{mnk} = \left< F_m \psi_i[\{\zeta_r\}] \right >
% \end{equation}
% \end{normalsize}
% \end{flushright}
\vspace*{0.5cm}
\begin{equation}
\nonumber
K_{mn} = \int_D B_n \textcolor{mycolor}{\bf D} B_m dV \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ K'_{mnk} = \int_D B_n {\sqrt \lambda_k h_k} B_m dV
\end{equation}
\vspace*{1.0cm}
\begin{equation}
\nonumber
C_{ijk} = \left < \xi_k(\theta) \psi_i[\{\xi_r\}] \psi_j[\{\xi_r\}] \right > \ \ \ \ \ \ \ \ \ \ \ \ F_m = \int_D \phi N_m dV \ \ \ \ \ \ \ \ \ \ \ \
\end{equation}
%\item FokkerPlanckKolmogorov approach based probabilistic constitutive integration
% at Gauss integration points
\end{itemize}
% \noindent Salient Features:
% \begin{itemize}
%
% \item Efficient representation of input random fields into finite number of random
% variables using KLexpansion
%
% \item Representation of (unknown) solution random variables using polynomial chaos of
% (known) input random variables
%
% \item FokkerPlanckKolmogorov approach based probabilistic constitutive integration
% at Gauss integration points
%
% \end{itemize}
%
%% \end{itemize}
%
\end{frame}
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\begin{frame}
\frametitle{Inside SEPFEM}
\begin{itemize}
\item Stochastic elasticplastic (explicit) finite element computations
\vspace*{0.2cm}
\item FPK probabilistic constitutive integration at Gauss integration points
\vspace*{0.2cm}
\item Increase in (stochastic) dimensions (KL and PC) of the problem
\vspace*{0.2cm}
\item Development of the probabilistic elasticplastic stiffness tensor
\end{itemize}
% \noindent Salient Features:
% \begin{itemize}
%
% \item Efficient representation of input random fields into finite number of random
% variables using KLexpansion
%
% \item Representation of (unknown) solution random variables using polynomial chaos of
% (known) input random variables
%
% \item FokkerPlanckKolmogorov approach based probabilistic constitutive integration
% at Gauss integration points
%
% \end{itemize}
%
%% \end{itemize}
%
\end{frame}
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\section{Applications to Risk Analysis}
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\subsection{Seismic Wave Propagation Through Uncertain Soils}
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\begin{frame}
\frametitle{Risk Assessment Applications}
\begin{itemize}
%\vspace*{0.2cm}
\item Any problem
(${\bf M} \ddot{\bf u}
+
{\bf C} \dot{\bf u}
+
{\bf K} {\bf u}
=
{\bf F}
$)
with known
\begin{itemize}
\item PDFs of material parameters,
\item PDFs of loading
\end{itemize}
can be analyzed using PEP and SEPFEM to obtain PDFs of DOFs,
stress, strain...
%\vspace*{0.2cm}
\item PEP solution is second order accurate (exact mean and standard deviation)
%\vspace*{0.2cm}
\item SEPFEM solution (PDFs) can be made as accurate as need be
\item Tails of PDFs can than be used to develop accurate risk
\item Application to a realistic case of seismic wave propagation
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{"Uniform" CPT Site Data}
\vspace*{0.7cm}
%\begin{figure}
\begin{center}
\includegraphics[height=6.7cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/EastWestProfileEdited.pdf}
\end{center}
%\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Seismic Wave Propagation through Stochastic Soil}
%\begin{flushleft}
%\includegraphics[height=5.0cm]{PEER2007_3.jpg}
%\end{flushleft}
%\vspace*{0.5truecm}
\begin{itemize}
\item Soil as 12.5 m deep 1D soil column (von Mises Material)
\begin{itemize}
\item Properties (including testing uncertainty) obtained through random field modeling of CPT $q_T$
%
$\left = 4.99 ~MPa;~~Var[q_T] = 25.67 ~MPa^2; $\\
Cor. ~Length $[q_T] = 0.61 ~m; $ Testing~Error $= 2.78 ~MPa^2$
\end{itemize}
\vspace*{0.2cm}
\item $q_T$ was transformed to obtain $G$: ~~$G/(1\nu)~=~2.9q_T$
\begin{itemize}
\item Assumed transformation uncertainty = 5\%
%
$\left = 11.57MPa; Var[G] = 142.32 MPa^2$ \\
Cor.~Length $[G] = 0.61 m$
\end{itemize}
%\begin{center}
%\hspace*{1.7cm}
%\includegraphics[height=3.5cm]{Chapter9_Schematic.jpg}
%\hspace*{0.0cm}
%\includegraphics[height=3.5cm]{Chapter9_BaseDisplacement.jpg} \\
%\small{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Base Displacement}
%\end{center}
\vspace*{0.2cm}
\item Input motions: modified 1938 Imperial Valley
% \vspace*{0.2cm}
% \begin{center}
% \includegraphics[height=2.0cm]{Chapter9_BaseDisplacement.jpg}
% \end{center}
\end{itemize}
\end{frame}
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% %\begin{frame}
% %
% %
% %\frametitle{Random Field Parameters from Site Data}
% %
% %\begin{itemize}
% %
% %%\item maximizing the loglikelihood of observing the spatial data under assumed joined distribution (for finite
% %%scale model) or maximizing the loglikelihood of observing the periodogram estimates (for fractal model)
% %
% %\item Maximum likelihood estimates
% %
% %\vspace*{0.3truecm}
% %
% %%\begin{figure}
% %\begin{flushleft}
% %\hspace*{1.7cm}
% %\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/SamplingPlanEdited.jpg}
% %\hspace*{0.0cm}
% %\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalDataPlotBH1Edited.jpg} \\
% %\small{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Typical CPT $q_T$}
% %\end{flushleft}
% %%\end{figure}
% %
% %\vspace*{4.9truecm}
% %
% %%\begin{figure}
% %\begin{flushright}
% %\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FiniteScaleEdited.jpg} \\
% %\vspace*{0.01truecm}
% %\small{Finite Scale}
% %\end{flushright}
% %%\end{figure}
% %
% %\vspace*{0.02truecm}
% %
% %%\begin{figure}
% %\begin{flushright}
% %\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FractalEdited.jpg} \\
% %\small{Fractal}
% %\end{flushright}
% %%\end{figure}
% %
% %\end{itemize}
% %
% %\end{frame}
% %
% %
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% \subsection{Seismic Wave Propagation Through Uncertain Soils}
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\begin{frame}
\frametitle{Seismic Wave Propagation through Stochastic Soil}
\begin{figure}
\begin{center}
\hspace*{0.75cm}
\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter9Plots/Chapter9_ElasticPlasticResponseNew.pdf}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
Mean$\pm$ Standard Deviation
%\begin{flushleft}
%\includegraphics[height=5.0cm]{PEER2007_3.jpg}
%\end{flushleft}
% \hspace*{1.0cm} \noindent Statistics of Top Node Displacement:
%
% \vspace*{0.5truecm}
%
% \begin{figure}
% \begin{flushleft}
% \hspace*{1.0cm}
% \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_MeanNew.jpg}
% \hspace*{0.1cm}
% \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_SDNew.jpg}
% \end{flushleft}
% \end{figure}
% \vspace*{0.5truecm}
% \hspace*{1.0cm} \tiny{~~~~~~~~~~~~~~~~~~~~~~~~~~~~Mean~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Standard Deviation}
%
% \vspace*{0.3truecm}
%
% \begin{figure}
% \begin{flushleft}
% \hspace*{0.75cm}
% \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponseNew.jpg}
% \hspace*{0.4cm}
% \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_COVNew.jpg}
% \end{flushleft}
% \end{figure}
% \vspace*{0.3truecm}
% \hspace*{0.5cm} \tiny{~~~~~~~Mean$\pm$ Standard Deviation~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~COV}
%
%
% \vspace*{6.0cm}
% \begin{flushright}
% \includegraphics[height=4.5cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_PDFNewEdited.jpg} \hspace*{1.0cm}
% \end{flushright}
%
\end{frame}
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% %
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\begin{frame}
\frametitle{Full PDFs All Results}
\begin{figure}
\begin{center}
\vspace*{0.75cm}
\includegraphics[width=7.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/EvolutionaryPDF_ActualEdited.pdf}
\vspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
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\end{frame}
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\subsection{Probabilistic Analysis for Decision Making}
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\begin{frame}
\frametitle{Example: Three Approaches to Modeling}
\begin{itemize}
\vspace*{0.3cm}
\item {\bf Do nothing} about site (material) characterization (rely on
experience): conservative {guess} for soil data, $COV = 225$\%, correlation
length $= 12$m.
\vspace*{0.3cm}
\item {\bf Do better} than standard site (material) characterization: $COV =
103$\%, correlation length $= 0.61$m)
\vspace*{0.3cm}
\item {\bf Do the best} site (material) characterization to reduce
probabilities of exceedance
\end{itemize}
\end{frame}
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% %
% %\begin{frame}
% %
% %
% %\frametitle{Evolution of Mean $\pm$ SD for Guess Case}
% %
% %
% %
% %\begin{figure}
% %\begin{center}
% %\hspace*{0.75cm}
% %\includegraphics[width=10.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/Evolutionary_Mean_pm_SD_NoDataEdited.pdf}
% %\hspace*{0.75cm}
% %%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
% %\end{center}
% %\end{figure}
% %
% %%
% %\end{frame}
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% %\begin{frame}
% %
% %
% %\frametitle{Evolution of Mean $\pm$ SD for Real Data Case}
% %
% %
% %
% %\begin{figure}
% %\begin{center}
% %\hspace*{0.75cm}
% %\includegraphics[width=10.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/Evolutionary_Mean_pm_SD_ActualEdited.pdf}
% %\hspace*{0.75cm}
% %%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
% %\end{center}
% %\end{figure}
% %
% %%
% %\end{frame}
% %
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\begin{frame}
\frametitle{Example: PDF at $6$ s}
\begin{figure}
\begin{center}
\hspace*{1.75cm}
\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/PDFs_at6sec_Actual_vs_NoDataEdited.pdf}
\vspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
%
\end{frame}
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\begin{frame}
\frametitle{Example: CDF (NonExceedance) at $6$ s}
\begin{figure}
\begin{center}
%\hspace*{0.75cm}
\includegraphics[width=8.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/CDFs_at6sec_Actual_vs_NoDataEdited.pdf}
\vspace*{0.75cm}
%\hspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
%
\end{frame}
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\begin{frame}
\frametitle{Probability of Exceedance of $20$cm}
\begin{figure}
\begin{center}
%\hspace*{0.75cm}
\includegraphics[width=11.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/NewPlots/with_legends_and_labels/Exceedance20cm_LomaPrietaEdited.pdf}
\vspace*{0.75cm}
%\hspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Probability of Exceedance of $50$cm}
\begin{figure}
\begin{center}
%\hspace*{0.75cm}
\includegraphics[width=11.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/NewPlots/with_legends_and_labels/Exceedance50cm_LomaPrietaEdited.pdf}
%\vspace*{0.75cm}
%\hspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Risk of Unacceptable Deformation}
\begin{figure}
\begin{center}
%\hspace*{0.75cm}
\includegraphics[width=8.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/NewPlots/with_legends_and_labels/Summary_LomaPrietaEdited.pdf}
\vspace*{0.75cm}
%\hspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
\end{frame}
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\section{Summary}
\begin{frame}
\frametitle{Summary}
\begin{itemize}
\item Behavior of all civil engineering objects (structures, soils...) is
probably probabilistic!
\vspace*{0.3cm}
\item Presented methodology (PEP and SEPFEM) allows for (very) accurate
numerical simulation of PDFs of DOFs (and stress, strain) from
known (given) PDFs of material properties and PDFs of loads.
\vspace*{0.3cm}
\item Human nature: how much do you want to know about potential problem?
\end{itemize}
\end{frame}
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% 2000.
% \end{thebibliography}
%\end{frame}
\end{document}