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% % (used in a file _Chapter_SoftwareHardware_Domain_Specific_Language_English.tex
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% \usetheme{Antibes} % ima sadrzaj gore i kao graf ...
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% does not look nice, try deleting the line with the fontenc.
% Site Specific Dynamics of Structures:
%From Seismic Source to
%the Safety of Occupants and Content
\title[SEPFEM]
{Stochastic ElasticPlastic \\
Finite Element Method: \\
Recent Advances and Developments}
%\subtitle
%{Include Only If Paper Has a Subtitle}
%\author[Author, Another] % (optional, use only with lots of authors)
%{F.~Author\inst{1} \and S.~Another\inst{2}}
%  Give the names in the same order as the appear in the paper.
%  Use the \inst{?} command only if the authors have different
% affiliation.
\pgfdeclareimage[height=0.2cm]{universitylogo}{/home/jeremic/BG/amblemi/ucdavis_logo_blue_sm}
\pgfdeclareimage[height=0.7cm]{lbnllogo}{/home/jeremic/BG/amblemi/lbnllogo}
\author[Jeremi{\'c} and Lacour] % (optional, use only with lots of authors)
%{Boris~Jeremi{\'c}}
{Boris Jeremi{\'c} and Maxime Lacour
}
%\institute[Computational Geomechanics Group \hspace*{0.3truecm}
\institute[\pgfuseimage{universitylogo}\hspace*{0.1truecm}\pgfuseimage{lbnllogo}] % (optional, but mostly needed)
%{ Professor, University of California, Davis\\
{ University of California, Davis, CA\\
% and\\
% Faculty Scientist, Lawrence Berkeley National Laboratory, Berkeley }
Lawrence Berkeley National Laboratory, Berkeley, CA}
%  Use the \inst command only if there are several affiliations.
%  Keep it simple, no one is interested in your street address.
\date[] % (optional, should be abbreviation of conference name)
{\small SEECCM 2017\\ ~ \\
{\cyrdvanaest Kragujevac, Srbija} }
\subject{}
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\begin{frame}
\frametitle{Outline}
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% You might wish to add the option [pausesections]
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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).
%  At *most* three subsections per section.
%  Talk about 30s to 2min per frame. So there should be between about
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%  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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
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\subsection*{Motivation}
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\begin{frame}
\frametitle{Motivation}
\begin{itemize}
\item Probabilistic fish counting
\vspace*{3mm}
\item Williams' DEM simulations, differential displacement vortices
%\vspace*{0.2cm}
\vspace*{3mm}
\item SFEM round table
%\vspace*{0.2cm}
\vspace*{3mm}
\item Kavvas' probabilistic hydrology
\vspace*{3mm}
\item Performance based design, probability of undesirable performance,
($10^{4}$, $10^{5}$ !?)
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Material Behavior Inherently Uncertain}
\begin{itemize}
\vspace*{0.5cm}
\item Spatial \\
variability
\vspace*{0.5cm}
\item Pointwise \\
uncertainty, \\
testing \\
error, \\
transformation \\
error
\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}
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\begin{frame}
\frametitle{Parametric Uncertainty: Material and Loads}
%
% \begin{itemize}
%
%
%
% \item Significant uncertainty in material and loads
% %
% % %\vspace*{1mm}
% % \item Propagate uncertainties in space and time
% %
% %
% \end{itemize}
%
% %
%\vspace{1cm}
%\hspace{0.5cm}
%Example: Elastic Stiffness
%\vspace*{3mm}
\begin{figure}[!hbpt]
\begin{center}
%
\hspace*{7mm}
\includegraphics[width=7.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01Ed.pdf}
% \hfill
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_Histogram_Normal_01Ed.pdf}
%
\end{center}
\end{figure}
\vspace*{0.8cm}
%\hspace*{3.3cm}
\begin{flushleft}
{\tiny
Transformation of SPT $N$value:
1D Young's modulus, $E$
(cf. Phoon and Kulhawy (1999B))
~}
\end{flushleft}
\end{frame}
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\begin{frame}
\frametitle{Parametric Uncertainty: Material Properties}
\begin{figure}[!hbpt]
\begin{center}
% %
\hspace*{3mm}
\includegraphics[width=3.0truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldPhiPdf.pdf}
\hspace*{3mm}
\includegraphics[width=3.0truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldPhiCdf.pdf}
\hspace*{3mm}
\includegraphics[width=3.0truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldSuPdf.pdf}
\hspace*{3mm}
\includegraphics[width=3.0truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldSuCdf.pdf}
\\
%\vspace*{2mm}
\hspace*{2.5cm} \mbox{\tiny Field $\phi$} \hspace*{3.5cm} \mbox{\tiny Field $c_u$}
%\vspace*{45mm}
\hspace*{3mm}
\includegraphics[width=3.0truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabPhiPdf.pdf}
\hspace*{3mm}
\includegraphics[width=3.0truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabPhiCdf.pdf}
\hspace*{3mm}
\includegraphics[width=3.0truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabSuPdf.pdf}
\hspace*{3mm}
\includegraphics[width=3.0truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabSuCdf.pdf}
\\
%\vspace*{8mm}
\hspace*{2.5cm} \mbox{\tiny Lab $\phi$} \hspace*{3.5cm} \mbox{\tiny Lab $c_u$}
\end{center}
\end{figure}
\end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{It is not a new Problem}
\vspace*{10mm}
{\Large Le doute n'est pas un {\'e}tat bien agr{\'e}able,\\
mais l'assurance est un {\'e}tat ridicule.}
\vspace*{10mm}
\begin{flushright}
Fran{\c c}oisMarie Arouet (Voltaire)
\end{flushright}
\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}
\item Ergodic Assumption!?
\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}
\end{figure}
\end{frame}
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\section{Probabilistic Inelastic Modeling}
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\subsection{FPK Formulations}
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\begin{frame}
\frametitle{Uncertainty Propagation through
Inelastic System}
%
\begin{itemize}
\item Incremental elpl constitutive equation
%
\begin{eqnarray}
\nonumber
\Delta \sigma_{ij}
=
% E^{EP}_{ijkl}
E^{EP}_{ijkl} \; \Delta \epsilon_{kl}
=
\left[
E^{el}_{ijkl}

\frac{\displaystyle E^{el}_{ijmn} m_{mn} n_{pq} E^{el}_{pqkl}}
{\displaystyle n_{rs} E^{el}_{rstu} m_{tu}  \xi_* h_*}
\right]
\Delta \epsilon_{kl}
\end{eqnarray}
\vspace*{5mm}
\item Dynamic Finite Elements
%
\begin{equation}
{\bf M} \ddot{\bf u} +
{\bf C} \dot{\bf u} +
{\bf K}^{ep} {\bf u} =
{\bf F}
\nonumber
\end{equation}
\item What if all (any) material and load parameters are uncertain
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Probabilistic ElasticPlastic Response}
\begin{figure}[!hbpt]
\begin{center}
%\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDFedited.pdf}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2012/DOELLNLworkshop2728Feb2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_Contour_PDFedited.pdf}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Previous Work}
\begin{itemize}
\item
Linear algebraic or differential equations:
\begin{itemize}
\item Variable Transf. Method (Montgomery and Runger 2003)
\item Cumulant Expansion Method (Gardiner 2004)
\end{itemize}
\item
Nonlinear differential equations:
\begin{itemize}
\item Monte Carlo Simulation (Schueller 1997, De Lima et al 2001, Mellah
et al. 2000, Griffiths et al. 2005...) \\ $\rightarrow$ can be accurate, very costly
\item Perturbation Method (Anders and Hori 2000, Kleiber and Hien 1992,
Matthies et al. 1997) \\ $\rightarrow$ first and second order Taylor series
expansion about mean  limited to problems with small C.O.V. and inherits
"closure problem"
\item SFEM (Matthies et al, 2004, 2005, 2014...)
\end{itemize}
%
% \item
% Monte Carlo method: accurate, very costly
%
% \item
% Perturbation method:
\end{itemize}
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\end{frame}
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\begin{frame} \frametitle{{3D FPK Equation}}
\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}
% \begin{itemize}
%
%
%
% \item 6 equations
%
% \item Complete description of 3D probabilistic stressstrain response
%
% \end{itemize}
%
%
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\end{frame}
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\begin{frame}
\frametitle{FPK Equation}
\begin{itemize}
\item Advectiondiffusion equation
%
\begin{equation}
\nonumber
\frac{\partial P(\sigma,t)}{\partial t} = \frac{\partial}{\partial \sigma}\left[N_{(1)}P(\sigma,t)\frac{\partial}{\partial \sigma}
\left\{N_{(2)} P(\sigma,t)\right\} \right]
\end{equation}
%
\item Complete probabilistic description of response
\item Solution PDF is secondorder exact to covariance of time (exact mean and variance)
\item It is deterministic equation in probability density space
\item It is linear PDE in probability density space
$\rightarrow$ simplifies the numerical solution process
%\vspace*{0.2truecm}
\end{itemize}
%
% \vspace*{0.5cm}
% {%
% \begin{beamercolorbox}{section in head/foot}
% \usebeamerfont{framesubtitle}\tiny{B. Jeremi\'{c}, K. Sett, and M. L. Kavvas, "Probabilistic
% ElastoPlasticity: Formulation in 1D", \textit{Acta Geotechnica}, Vol. 2, No. 3, 2007, In press (published
% online in the \textit{Online First} section)}
% %\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
% \end{beamercolorbox}%
% }
\end{frame}
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\begin{frame}
\frametitle{Template Solution of FPK Equation}
\begin{itemize}
\item FPK diffusionadvection equation is applicable to any material model $\rightarrow$
only the coefficients $N_{(1)}$ and $N_{(2)}$ are different for different material models
% %
% %
% %
% %\begin{normalsize}
% \begin{equation}
% \nonumber
% \frac{\partial P(\sigma,t)}{\partial t} = \frac{\partial}{\partial \sigma}\left[N_{(1)}P(\sigma,t)\frac{\partial}{\partial \sigma}
% \left\{N_{(2)} P(\sigma,t)\right\} \right]
% %\nonumber
% = \frac{\partial \zeta}{\partial \sigma}
% \end{equation}
% %\end{normalsize}
%
\item Initial condition
\begin{itemize}
\item Deterministic $\rightarrow$ Dirac delta function $\rightarrow$ $ P(\sigma,0)=\delta(\sigma) $
\item Random $\rightarrow$ Any given distribution
\end{itemize}
\item Boundary condition: Reflecting BC $\rightarrow$ conserves probability mass
$\zeta(\sigma,t)_{At \ Boundaries}=0$
\item Solve using finite differences and/or finite elements
\item However (!!) it is a stress solution and probabilistic stiffness is an
approximation!
\end{itemize}
\end{frame}
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\subsection{Direct Solution for Probabilistic Stiffness and Stress}
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\begin{frame}{Direct Probabilistic Constitutive Modeling}
% \begin{itemize}
%
% \vspace{0.5cm}
%
% \item<1> Probabilistic constitutive modeling : \vspace{0.5cm}
\begin{itemize}
\item Zero elastic region elastoplasticity with stochastic ArmstrongFrederick
kinematic hardening
$ \Delta\sigma =\ H_a \Delta \epsilon  c_r \sigma \Delta \epsilon ;
\hspace{0.5cm}
E_t = {d\sigma}/{d\epsilon} = H_a \pm c_r \sigma $
\vspace*{2mm}
\item Uncertain:
init. stiff. $H_a$,
shear strength $H_a/c_r$,
strain $\Delta \epsilon$:
$ H_a = \Sigma h_i \Phi_i; \;\;\;
C_r = \Sigma c_i \Phi_i; \;\;\;
\Delta\epsilon = \Sigma \Delta\epsilon_i \Phi_i $
\vspace*{2mm}
\item Resulting stress and stiffness are also uncertain
% 
%  $ \sum_{l=1}^{P_{\sigma}} \Delta\sigma_i \Phi_i = \sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k \Phi_i \Phi_k  \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_i \Delta \epsilon_k \sigma_l \Phi_j \Phi_k \Phi_l$
% 
%  $ \sum_{l=1}^{P_{E_t}} \Delta E_{t_i} \Phi_i = \sum_{i=1}^{P_h} h_i \Phi_i \pm \sum_{i=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_i \sigma_l \Phi_i \Phi_l$
% 
\end{itemize}
% \vspace{0.5cm}
% \vspace{1cm}
%\item<1> Time integration is done via Newmark algorithm
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% \end{itemize}
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\end{frame}
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\begin{frame}{Probabilistic Stiffness Solution}
\begin{itemize}
\item Analytic product for all the components,
$ E^{EP}_{ijkl}
=
\left[
E^{el}_{ijkl}

\frac{\displaystyle E^{el}_{ijmn} m_{mn} n_{pq} E^{el}_{pqkl}}
{\displaystyle n_{rs} E^{el}_{rstu} m_{tu}  \xi_* h_*}
\right]
$
\item Stiffness: each Polynomial Chaos component is updated incrementally
% at each Gauss Point via stochastic Galerkin projection
\small{$E_{t_1}^{n+1} = \frac{1}{<\Phi_1\Phi_1> }\{\sum_{i=1}^{P_h} \ h_i <\Phi_i \Phi_1> \pm \sum_{j=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_j \sigma_l^{n+1} <\Phi_j \Phi_l \Phi_1>\}$}
$\large{\vdots}$
\small{$E_{t_P}^{n+1} = \frac{1}{<\Phi_1\Phi_P> }\{\sum_{i=1}^{P_h} \ h_i <\Phi_i \Phi_P> \pm \sum_{j=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_j \sigma_l^{n+1} <\Phi_j \Phi_l \Phi_P>\}$}
\item Total stiffness is :
$ E_{t}^{n+1} = \sum_{l=1}^{P_{E}} E_{t_i}^{n+1} \Phi_i $
\end{itemize}
\end{frame}
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\begin{frame}{Probabilistic Stress Solution}
\begin{itemize}
\item Analytic product, for each stress component,
$ \Delta \sigma_{ij} = E^{EP}_{ijkl} \; \Delta \epsilon_{kl} $
% =
% \left[
% E^{el}_{ijkl}
% 
% \frac{\displaystyle E^{el}_{ijmn} m_{mn} n_{pq} E^{el}_{pqkl}}
% {\displaystyle n_{rs} E^{el}_{rstu} m_{tu}  \xi_* h_*}
% \right]
% \Delta \epsilon_{kl}
%
\vspace*{1mm}
\item Incremental stress: each Polynomial Chaos component is updated
incrementally
% via stochastic Galerkin projection
{$\Delta\sigma_1^{n+1} = \frac{1}{<\Phi_1\Phi_1> }\{\sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k^n <\Phi_i \Phi_k \Phi_1> \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_j \Delta \epsilon_k^n \sigma_l^n <\Phi_j \Phi_k \Phi_l \Phi_1>\}$}
${\vdots}$
{$\Delta\sigma_P^{n+1} = \frac{1}{<\Phi_P\Phi_P> }\{\sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k^n <\Phi_i \Phi_k \Phi_P> \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_j \Delta \epsilon_k^n \sigma_l^n <\Phi_j \Phi_k \Phi_l \Phi_P>\}$}
\vspace*{1mm}
\item Stress update:
$ \sum_{l=1}^{P_{\sigma}} \sigma_i^{n+1} \Phi_i = \sum_{l=1}^{P_{\sigma}} \sigma_i^{n} \Phi_i + \sum_{l=1}^{P_{\sigma}} \Delta\sigma_i^{n+1} \Phi_i$
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Probabilistic ElasticPlastic Modeling}
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% {/home/jeremic/tex/works/Thesis/HanYang/Files_06June2017/DOE_Annual_2017/Figures/NPP_Plastic_Dissipation_Density.png}}
% %{/home/jeremic/tex/works/Thesis/HanYang/Files_06June2017/DOE_Annual_2017/Figures/NPP_without_Contact_vonMises.mp4}
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\begin{center}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/PEP_Animation.mp4}
{\includegraphics[width=90mm]{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/figure_PEP_25.png}}
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\begin{frame}
\frametitle{Stochastic ElasticPlastic Finite Element Method (SEPFEM)}
\begin{itemize}
\item[] Material uncertainty expanded along stochastic shape functions:
$E(x,t,\theta) = \sum_{i=0}^{P_d} r_i(x,t) * \Phi_i[\{\xi_1, ..., \xi_m\}]$
\vspace*{4mm}
\item[] Loading uncertainty expanded along stochastic shape functions:
$f(x,t,\theta) = \sum_{i=0}^{P_f} f_i(x,t) * \zeta_i[\{\xi_{m+1}, ..., \xi_f]$
\vspace*{4mm}
\item[] Displacement expanded along stochastic shape functions:
$u(x,t,\theta) = \sum_{i=0}^{P_u} u_i(x,t) * \Psi_i[\{\xi_1, ..., \xi_m, \xi_{m+1}, ..., \xi_f\}]$
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{SEPFEM: Formulation}
\begin{itemize}
\item
Stochastic system of equation resulting from Galerkin approach (static example):
\begin{tiny}
\[
%$
\begin{bmatrix}
\sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
\sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\ \\
\vdots & \vdots & \vdots & \vdots\\
\sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)} & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
\end{bmatrix}
\begin{bmatrix}
\Delta u_{10} \\
\vdots \\
\Delta u_{N0}\\
\vdots \\
\Delta u_{1P_u}\\
\vdots \\
\Delta u_{NP_u}
\end{bmatrix}
=
%\]
%\[
\begin{bmatrix}
\sum_{i=0}^{P_f} f_i <\Psi_0\zeta_i> \\
\sum_{i=0}^{P_f} f_i <\Psi_1\zeta_i> \\
\sum_{i=0}^{P_f} f_i <\Psi_2\zeta_i> \\
\vdots \\
\sum_{i=0}^{P_f} f_i <\Psi_{P_u}\zeta_i>\\
\end{bmatrix}
%$
\]
\end{tiny}
\item Time domain integration using Newmark and/or HHT, in probabilistic spaces
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{SEPFEM: System Size}
\begin{itemize}
\item SEPFEM offers a complete solution (single step)
\item It is not based on Monte Carlo approach
\item System of equations does grow (!)
\end{itemize}
% \normalsize{Typical number of terms required for a SEPFEM problem} \vspace{1cm}\\
\scalebox{0.7}{
\begin{tabular}{ c c c c}
\# KL terms material & \# KL terms load & PC order displacement& Total \# terms per DoF\\ \hline
4 & 4 & 10 & 43758 \\
4 & 4 & 20 & 3 108 105 \\
4 & 4 & 30 & 48 903 492 \\
6 & 6 & 10 & 646 646 \\
6 & 6 & 20 & 225 792 840 \\
6 & 6 & 30 & 1.1058 $10^{10}$ \\
8 & 8 & 10 & 5 311 735 \\
8 & 8 & 20 & 7.3079 $10^{9}$ \\
8 & 8 & 30 & 9.9149 $10^{11}$\\
... & ... & ... & ...\\
% \hline
\end{tabular}}
\end{frame}
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\begin{frame}
\frametitle{SEPFEM: Example in 1D}
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% local \movie[label=show3,width=9cm,poster,autostart,showcontrols]
% local {\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_elastic_900.png}}
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\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/SEPFEM_Animation_Elastic.mp4}
{\includegraphics[width=90mm]{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/figure_elastic_900.png}}
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\begin{frame}
\frametitle{SEPFEM: Example in 3D}
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% local % \hspace*{15mm}
% local \movie[label=show3,width=11cm,poster,autostart,showcontrols]
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% local {/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/SFEM_3D.png}}
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% local % \includegraphics[width = 12cm]{./img/SFEM_3D.pdf}
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\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/SFEM_Animation_3D.mp4}
{\includegraphics[width=90mm]{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/SFEM_3D.png}}
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\begin{frame}{Application of SEPFEM to Practical Problems}
Obtain accurate fragility curves (CDFs) for each soil structure system location
for stress, strain, displacements, etc.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=3.5cm]{/home/jeremic/tex/works/Conferences/2017/SEECMM_2017_Kragujevac/present/Brana_3D_01.jpg}
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\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
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\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Conferences/2017/SEECMM_2017_Kragujevac/present/NPP01.jpg}
%\includegraphics[width=5.0cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Applications_Slope_Stability_in_2D_and_3D/3D_final04.jpg}
%//
%\includegraphics[width=5.0cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Applications_Slope_Stability_in_2D_and_3D/3D_final05.jpg}
%\includegraphics[width=5.0cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Applications_Slope_Stability_in_2D_and_3D/3D_final_Top.jpg}
%\hspace*{0.9cm}
%bridge.}
\end{center}
\end{figure}
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% \includegraphics[width=5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_NEES_Bridge/tex_works_Thesis_GuanzhouJie_thesis_Verzija_Februar_Images_3BentModel.pdf}
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\section{Summary}
\subsection*{Summary}
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% \begin{frame}
% \frametitle{USDOE Project for ESSI of Nuclear Facilities}
%
% %OVDE
% \begin{itemize}
%
% \item[] The Real ESSI Simulator (inelastic, deterministic and probabilistic, time domain, 3D FEM)
% % \href{http://realessi.us}{http://realessi.us}
%
% % \vspace*{2mm}
% \item[] Modeling from seismic source to NPP (SW4, Real ESSI)
%
% % \vspace*{2mm}
% \item[] Extensive Verification (NQA1, ISO), and Validation
%
%
%
% \vspace*{2mm}
% \begin{figure}[!hbpt]
% \begin{center}
% %
% \hspace*{7mm}
% \includegraphics[width=8.2truecm]{/home/jeremic/tex/works/Conferences/2017/CNWG_INL_1618_May_2017/Presentation/DOE_project_UNR_test_01.jpg}
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\begin{frame}
\frametitle{Summary}
%OVDE
\begin{itemize}
\item[] Probable importance of uncertainty in mechanics
\vspace*{2mm}
\item[] Propagation of uncertainty through mechanics in order to give
designers, regulators and users information for decision making
\vspace*{2mm}
\item[] Real ESSI Simulator:
\href{http://realessi.us}{http://realessi.us}
% \vspace*{1mm}
% \item[] Collaborators: Feng, Lacour, Han, Behbehani, Sinha, Wang,
% Pisan{\'o}, Abell, McCallen, McKenna, Petrone, Rodgers, Petersson, Pitarka
\vspace*{2mm}
\item[] Funding from and collaboration with the USDOE, USNRC, USNSF,
CNSCCCSN, UNIAEA, and Shimizu Corp. is greatly appreciated,
\end{itemize}
\end{frame}
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