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\title[Dynamics of Soils and Structures under Uncertainty]
{Dynamics of Soils and Structures under Uncertainty}
%\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.
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\author[Jeremi{\'c} et al.] % (optional, use only with lots of authors)
%{Boris~Jeremi{\'c}}
{Boris Jeremi{\'c} \\
Kallol Sett (UB), Konstantinos Karapiperis and Jos{\' e} Abell }
%\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\\
% and\\
% Faculty Scientist, Lawrence Berkeley National Laboratory, Berkeley }
Lawrence Berkeley National Laboratory, Berkeley }
%  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 CompDyn, \\ Crete, Greece, May 2015}
\subject{}
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\begin{frame}
\frametitle{Outline}
\begin{scriptsize}
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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
% 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.
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\section{Motivation}
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\subsection*{Introduction}
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\begin{frame}
\frametitle{Motivation}
\begin{itemize}
%\vspace*{0.3cm}
\item Improve seismic design of soil structure systems
\vspace*{1mm}
\item {E}arthquake {S}oil {S}tructure {I}nteraction
({ESSI}) in time and space, plays a major role in successes and failures
%\vspace*{0.1cm}
\vspace*{1mm}
\item Accurate following and directing (!) the flow of seismic energy in
ESSI system to optimize for
\begin{itemize}
\item Safety and
\item Economy
\end{itemize}
%\vspace*{0.1cm}
\vspace*{1mm}
\item Development of high fidelity numerical modeling and simulation tools
to analyze realistic ESSI behavior: \\
{Real ESSI} simulator
% ({\small aka}: {\cyrssDvanaest Stvarno Lako},
% { Muy F{\'a}cil},
% {Molto Facile},
% \raisebox{1.2mm}{\includegraphics[height=5mm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/Real_ESSI_in_different_langauges/Real_ESSI_Chinese.jpeg}},
% \raisebox{1.2mm}{\includegraphics[height=5mm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/Real_ESSI_in_different_langauges/Real_ESSI_Japanese.jpg}},
% {\greektext{Pragmatik'a E'ukolo}},
% \raisebox{1.2mm}{\includegraphics[height=5mm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/Real_ESSI_in_different_langauges/Real_ESSI_Farsi.jpg}},
% {Tr{\`e}s Facile},
% {\cyrssDvanaest Vistinski Lesno}
% %{Wirklich Einfach}
% )
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Predictive Capabilities}
% \frametitle{High Fidelity Modeling of SFS System:
% Verification, Validation and Prediction}
\begin{itemize}
\item {{ Verification} provides evidence that the model is solved
correctly.} Mathematics issue.
%\vspace*{0.1cm}
\vspace*{1mm}
\item {{ Validation} provides evidence that the correct model is
solved.} Physics issue.
%\vspace*{0.1cm}
\vspace*{1mm}
\item { Prediction under Uncertainty (!)}: use of computational model
to foretell the state of a physical system under consideration under
conditions for which the computational model has not been validated.
\vspace*{1mm}
\item Modeling and Parametric Uncertainties
\vspace*{1mm}
\item Predictive capabilities with {low Kolmogorov Complexity}
\vspace*{1mm}
\item Modeling and simulation goal is to inform, not fit
%
\end{itemize}
\end{frame}
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\subsection*{Uncertainties}
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\begin{frame}
\frametitle{Modeling Uncertainty}
\begin{itemize}
\item Simplified modeling: Features (important ?) are neglected (6D
ground motions, inelasticity)
\vspace*{4mm}
\item Modeling Uncertainty: unrealistic and unnecessary modeling
simplifications
\vspace*{4mm}
\item Modeling simplifications are justifiable if one or two level higher
sophistication model shows that features being simplified out are not
important
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Parametric Uncertainty: Material 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*{1.8cm}
% %\hspace*{3.3cm}
% \begin{flushright}
% {\tiny
% Transformation of SPT $N$value: \\
% 1D Young's modulus, $E$ \\
% (cf. Phoon and Kulhawy (1999B))\\
% ~}
% \end{flushright}
%
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Parametric Uncertainty: Material Strength}
%
%
% \begin{figure}[!hbpt]
% \begin{center}
% %
% \hspace*{7mm}
% \includegraphics[width=6.50truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/ShearStrength_RawData_and_MeanTrendMod.pdf}
% \hspace*{7mm}
% % \hfill
% \includegraphics[width=6.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/ShearStrength_Histogram_PearsonIVFineTunedMod.pdf}
% %
% \end{center}
% \end{figure}
%
% % \vspace*{1.8cm}
% % %\hspace*{3.3cm}
% % \begin{flushright}
% % {\tiny
% % Transformation of SPT $N$value: \\
% % 1D Young's modulus, $E$ \\
% % (cf. Phoon and Kulhawy (1999B))\\
% % ~}
% % \end{flushright}
% % %
%
%
% \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{Realistic ESSI Modeling Uncertainties}
\begin{itemize}
\item Seismic Motions: 6D, inclined, body and surface waves
(translations, rotations); Incoherency
\vspace*{3mm}
\item Inelastic material: soil, rock, concrete, steel; Contacts,
foundationsoil, dry, saturated slipgap; Nonlinear buoyant forces; Isolators,
Dissipators
\vspace*{3mm}
\item Uncertain loading and material
\vspace*{3mm}
\item Verification and Validation $\Rightarrow$ Predictions
% \vspace*{2mm}
% \item High Fidelity Models $\Rightarrow$ High Performance Computing
%
%
% \vspace*{2mm}
% \item Education
%
\end{itemize}
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
%
%
%
%
%
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\section{Modeling and Parametric Uncertainty}
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\subsection{Modeling Uncertainty}
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\begin{frame}
\frametitle{Real ESSI Models}
\begin{itemize}
\item Full seismic motion input (body and surface waves) using the Domain
Reduction Method (Bielak et al.)
\vspace*{2mm}
\item Inelastic (saturated or dry) soil/rock
\vspace*{2mm}
\item Inelastic (saturated or dry) contact (foundation  soil/rock)
\vspace*{2mm}
\item Buoyant (nonlinear) forces
\vspace*{2mm}
\item Inelastic structural modeling (elastic requested?!)
\vspace*{2mm}
\item Verification (extensive) and Validation (in progress)
\end{itemize}
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{6D Free Field Motions}
\vspace*{2mm}
\begin{center}
%\movie[label=show3,width=8.5cm,poster,autostart,showcontrols,loop]
\hspace*{7mm}
\movie[label=show3,width=8.8cm,autostart,showcontrols]
{\includegraphics[width=70mm]{BJicon.png}}{movie_input.mp4}
\end{center}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_01AApr2015/movie_input.mp4}
% \href{./homo_50mmesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
%
\end{flushleft}
\end{frame}
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\begin{frame}
\frametitle{6D Free Field Motions (closeup)}
\vspace*{2mm}
\begin{center}
\hspace*{7mm}
\movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
{\includegraphics[width=70mm]{BJicon.png}}{movie_input_closeup.mp4}
\end{center}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_01AApr2015/movie_input_closeup.mp4}
% \href{./homo_50mmesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
\end{flushleft}
%
\end{frame}
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\begin{frame}
\frametitle{6D Free Field at Location}
\vspace*{2mm}
\begin{center}
\hspace*{7mm}
\movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
{\includegraphics[width=69mm]{BJicon.png}}{movie_ff_3d.mp4}
\end{center}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_3d.mp4}
% \href{./homo_50mmesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
\end{flushleft}
%
\end{frame}
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\begin{frame}
\frametitle{6D Earthquake Soil Structure Interaction}
\vspace*{2mm}
\begin{center}
\hspace*{7mm}
\movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
{\includegraphics[width=70mm]{BJicon.png}}
{movie_npp_3d.mp4}
\end{center}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_npp_3d.mp4}
{\tiny (MP4)}
\end{flushleft}
%
\end{frame}
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\begin{frame}
\frametitle{From 6D to 1D?}
\begin{itemize}
\item Assume that a full 6D (3D) motions at the surface are only recorded in one
horizontal direction
\item From such recorded motions one can develop a vertically propagating shear
wave in 1D
\item Apply such vertically propagating shear wave to the same soilstructure
system
\end{itemize}
\vspace*{3mm}
\begin{figure}[!H]
\begin{center}
\includegraphics[width=6.5cm]{/home/jeremic/tex/works/Conferences/2015/CompDyn/Present/6D_to_1D_01.jpg}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{1D Free Field at Location}
\vspace*{2mm}
\begin{center}
\hspace*{7mm}
\movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
{\includegraphics[width=70mm]{BJicon.png}}
{movie_ff_1d.mp4}
\end{center}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_1d.mp4}
{\tiny (MP4)}
\end{flushleft}
%
\end{frame}
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\begin{frame}
\frametitle{1D ESSI of NPP}
\vspace*{2mm}
\begin{center}
\hspace*{7mm}
\movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
{\includegraphics[width=70mm]{BJicon.png}}
{movie_npp_1d.mp4}
\end{center}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_npp_1d.mp4}
{\tiny (MP4)}
\end{flushleft}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{6D vs 1D NPP ESSI Response Comparison}
\vspace*{2mm}
\begin{center}
\hspace*{7mm}
\movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
{\includegraphics[width=69mm]{BJicon.png}}
{movie_2_npps.mp4}
\end{center}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_2_npps.mp4}
{\tiny (MP4)}
\end{flushleft}
%
\end{frame}
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% %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{6D vs 1D: Containment Displacement Response}
%
%
% \begin{figure}[!hp]
% \begin{center}
% \hspace*{13mm}
% \includegraphics[width=7cm]{/home/jeremic/tex/works/Thesis/JoseAntonioAbellMena/NPP_movies_19May2015/01_displacements_Containment_building_bottom.pdf}
% \hspace*{6mm}
% \includegraphics[width=7cm]{/home/jeremic/tex/works/Thesis/JoseAntonioAbellMena/NPP_movies_19May2015/02_displacements_Containment_building_top.pdf}
% \hspace*{10mm}
% \end{center}
% \end{figure}
%
%
%
% \end{frame}
%
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\begin{frame}
\frametitle{6D vs 1D: Containment Acceleration Response}
\begin{figure}[!hp]
\begin{center}
\hspace*{13mm}
\includegraphics[width=7cm]{/home/jeremic/tex/works/Thesis/JoseAntonioAbellMena/NPP_movies_19May2015/01_accelerations_Containment_building_bottom.pdf}
\hspace*{6mm}
\includegraphics[width=7cm]{/home/jeremic/tex/works/Thesis/JoseAntonioAbellMena/NPP_movies_19May2015/02_accelerations_Containment_building_top.pdf}
\hspace*{10mm}
\end{center}
\end{figure}
\end{frame}
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%\subsection{Parametric Uncertainty: Probabilistic Inelasticity}
\subsection{Parametric Uncertainty}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Uncertain Material Parameters and Loads}
\begin{itemize}
\item Decide on modeling complexity
\vspace*{3mm}
\item Determine model/material parameters
\vspace*{3mm}
\item Model/material parameters are uncertain!
\begin{itemize}
\item Measurements
\item Transformation
\item Spatial variability
\end {itemize}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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}
=
\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}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Critique of our Previous Work, PEP and SEPFEM}
\begin{itemize}
\item Constitutive weighted coefficients $N_1$ and $N_2$ do not work well for
stress solution!
\vspace{1mm}
\item We suggested that $\sigma(t)$ be considered a $\delta$correlated, and
based on that simplified stiffness equations. Both the assumption and the
resulting equation were not right.
\vspace{1mm}
\item On a SEPFEM level, stiffness needs update basis functions and KL
coefficients in each step. We updated the eigenvalues
$\lambda_i$ and kept the same structure (KarhunenLoeve) in the approximation
of the stiffness, which is not physical
\vspace{1mm}
\item Implicitly assumed that the stiffness remains Gaussian, which is
not the case
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Gradient Flow Theory of Probabilistic ElastoPlasticity}
Decomposition of an elastoplastic random process:
% into
%the following general form:
%
\begin{eqnarray}
\begin{array}{c c c c c}
\bigg( \displaystyle{\pder[]{t}}  \mathcal{L}_{rev} \bigg) P(\boldsymbol{\sigma},t) &=& 0& \qquad &\text{if }
\boldsymbol{\sigma} \in \Omega^{el}
\\
~
\\
\bigg( \displaystyle{\pder[]{t}}  \mathcal{L}_{irr} \bigg) P(\boldsymbol{\sigma},t) &=& 0& \qquad &\text{if }
\boldsymbol{\sigma} \in \Omega^{el} \cup \Omega^{pl}
\end{array}
\nonumber
\end{eqnarray}
%
\begin{itemize}
\item Reversible ($\mathcal{L}_{rev} $) and Irreversible ($\mathcal{L}_{irr} $) operators
%
\vspace*{2mm}
\item Yield PDF is an attractor, similar to plastic corrector
\vspace*{2mm}
\item Ergodicity of the elasticplastic process can be proven!
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Gradient Flow Theory of Probabilistic ElastoPlasticity}
\begin{itemize}
\item
Elastic, reversible process, FokkerPlanck (forward Kolmogorov) equation
%
\begin{equation}
\pder[P (\boldsymbol{\sigma},t)]{t}
= \nabla \cdot (\langle \boldsymbol{E} \dot{\boldsymbol{\epsilon}} \rangle P(\boldsymbol{\sigma},t))
+ t ~\text{Var}[\boldsymbol{E}\dot{\boldsymbol{\epsilon}}] \Delta P(\boldsymbol{\sigma},t)
\nonumber
\end{equation}
%
\begin{equation}
\mathcal{L}_{rev} = \nabla \cdot (t \,\text{Var}[\boldsymbol{C}\dot{\boldsymbol{\epsilon}}] \nabla 
\langle \boldsymbol{C} \dot{\boldsymbol{\epsilon}} \rangle)
\nonumber
\end{equation}
\item
Plastic, irreversible process, FokkerPlanck (forward Kolmogorov) equation
%
\begin{equation}
\pder[P(\boldsymbol{\sigma},t)]{t} = \nabla \cdot (\nabla \Psi(\boldsymbol{\sigma})
P(\boldsymbol{\sigma},t)) + D \Delta P(\boldsymbol{\sigma},t)
\nonumber
\end{equation}
%
\begin{equation}
\mathcal{L}_{irr} = D \nabla \cdot \bigg(\nabla  \frac{\nabla P_y(\boldsymbol{\sigma})}{P_y(\boldsymbol{\sigma})}\bigg)
\nonumber
\end{equation}
%
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Gradient Flow Theory of Probabilistic ElastoPlasticity}
\begin{itemize}
\item Limiting (final) distribution
is considered to be known
%(given as a material parameter) and
\vspace*{2mm}
\item Underlying potential leading to this distribution is sought
\vspace*{2mm}
\item Transition from uncertain elastic to uncertain plastic response
\vspace*{2mm}
\item Only in a 1D elastoplastic problem does one end up with a stationary
distribution
\vspace*{2mm}
\item In higher dimensional problems this yield stress distribution is
only "marginally" stationary along one or a combination of the stress
components.
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Probabilistic ElastoPlasticity: von Mises Surface}
% An example of a Mises probabilistic yield surface is given in Fig.~\ref{ProbCylinder}.
\begin{figure}[H]
\centering
\vspace*{5mm}
%\hspace*{30mm}
\includegraphics
% [scale = 0.8, trim = 0.6in 0.4in 0.6in 0.4in, clip=true]
[width=12cm]
{/home/jeremic/tex/works/Reports/2015/SEPFEM/figures/Probabilistic_von_Mises_Surface.jpg}
% \caption{An example of a probabilistic von Mises yield surface.}
\label{ProbCylinder}
\end{figure}
\end{frame}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Gradient Theory of Probabilistic ElastoPlasticity: Numerical Solution}
\begin{itemize}
\item Using radial basis functions (a meshless method) for solving
FokkerPlanck equations for uncertain elasticplastic response
\vspace*{6mm}
\item Details in a talk by Mr. Karapiperis later this afternoon (room 2, MS6,
17:0019:00, last talk)
\end{itemize}
% %
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Gradient Theory of Probabilistic ElastoPlasticity:
Verification, ElasticPerfectly Plastic}
%\vspace*{3mm}
\begin{figure}[H]
\centering
\includegraphics
% [scale = 0.55, trim = 0.0in 0.0in 0.0in 0.0in, clip=true]
[width=8cm]
{/home/jeremic/tex/works/Reports/2015/SEPFEM/figures/M+SD_1D_Perf_Cyclic.pdf}
% \caption{Combining the above plots to get the evolution of mean $\pm$ std.
% deviation of stress.}
\label{MSD_12}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{SEPFEM}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Stochastic ElasticPlastic FEM (SEPFEM)}
\begin{itemize}
\item<1> KLPC expansion of material random fields (stiffness, etc)
%
% \begin{equation*}
$
\mathbb{D}(\bold{x}, \theta) = \sum_{i=0}^{M} r_{i}(\bold{x}) \Phi_i[\{\xi_r(\theta)\}]
\label{nonGaussian1}
$
%\end{equation*}
\item<1> PC expansion of displacement field
%
\begin{eqnarray*}
u(\bold{x},\theta) = \sum_{i=0}^p d_i(\bold{x}) \psi_i[\{\xi_r(\theta)\}]
\label{PC1}
\end{eqnarray*}
\item<1> Stochastic Galerkin
\begin{eqnarray*}
\sum_{n = 1} ^ N K'_{mn} d_{ni} + \sum_{n=1}^N \sum_{j=0}^P d_{nj} \sum_{k=1}^M b_{ijk} K''_{mnk} =
\Phi_{m}\langle\psi_i[\{\xi_r\}]\rangle
\label{Lognormal6}
\end{eqnarray*}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{SEPFEM Statistical linearization}
Update the FE stiffness in the elastoplastic regime:
\begin{itemize}
\item Solve elastoplastic FPE for each integration point
%
\begin{eqnarray*}
\pder[P^{nl}(\sigma,t)]{t}
&=& \pder[]{\sigma} \bigg(\bigg\langle D^k (1  P[\Sigma_y \leq \sigma]) \frac{\Delta
\epsilon}{\Delta t} \bigg\rangle P\bigg) \nonumber \\
& & + \pder[^2]{\sigma^2} \bigg( t Var\bigg[ D^k (1  P[\Sigma_y \leq \sigma]) \frac{\Delta
\epsilon}{\Delta t} \bigg] P\bigg)
\label{SEPFEMLinearization1}
\end{eqnarray*}
\item Consider an equivalent linear FPE
%
\begin{eqnarray*}
\pder[P^{lin}(\sigma,t)]{t}
&=& N_{(1)}^{eq} \pder[P]{\sigma} + N_{(2)}^{eq} \pder[^2 P]{\sigma^2}
\label{SEPFEMLinearization2}
\end{eqnarray*}
\item Linearization of the PC coeff. as an optimization problem
%
\begin{equation*}
\pder[P^{lin}(\sigma,t)]{t}
= \pder[P^{nl}(\sigma,t)]{t}
\end{equation*}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Dynamic, Time Domain, SEPFEM}
\begin{itemize}
\item Gaussian formulation inadequate due to occurrence of "probabilistic softening" modes
$\Rightarrow$ Need for positive definite kernel
\vspace*{2mm}
\item Stochastic forcing (e.g. uncertain earthquake)
\vspace*{2mm}
\item Stability of time marching algorithm (Newmark, Rosenbrock, Cubic Hermitian ) analyzed using
amplification matrix
\vspace*{2mm}
\item Longintegration error and higher order statistics phase shift
\end{itemize}
% \begin{figure}[H]
% \centering
% \includegraphics[scale = 0.4, trim = 0.1in 0.0in 0.0in 0.0in, clip=true]{/home/jeremic/tex/Classes/2015/spring/ECI280A/SEPFEM/odKonstantinosa/figures/Sendai_1Sine_PCk2PCd8.pdf}
% \label{Sendai_1Sine_PCk2PCd8.pdf}
% \end{figure}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Wave Propagation Through Uncertain Soil}
\vspace{2mm}
\begin{figure}[H]
\centering
\includegraphics
% [scale = 0.56, trim = 0.0in 0.15in 0.2in 0.16in, clip=true]
[width=7cm]
{/home/jeremic/tex/works/Reports/2015/SEPFEM/figures/Sendai_profile.png}
%
%
\includegraphics
%[scale = 0.56, trim = 0.0in 0.0in 0.0in 0.0in, clip=true]
[width=1.5cm]
{/home/jeremic/tex/works/Reports/2015/SEPFEM/figures/meshes.png}
% \caption{Simplified profile and properties for the 1dimensional profile considered.}
\label{Sendai_profile}
\end{figure}
\vspace{5mm}
\begin{figure}[htbp]
\centering
%\hspace{0.2in}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\includegraphics
% [scale = 0.37, trim = 0.0in 0.0in 0.0in 0.0in,clip=true]
[width=3.5cm]
{/home/jeremic/tex/works/Reports/2015/SEPFEM/figures/ShearModulusStats.pdf}
\hfill
\includegraphics
% [scale = 0.37, trim = 0.0in 0.0in 0.0in 0.0in, clip=true]
[width=3.5cm]
{/home/jeremic/tex/works/Reports/2015/SEPFEM/figures/ShearStrengthStats.pdf}
%\caption{Shear modulus and shear strength statistics in the middle of the soil column.}
\hfill
\includegraphics
% [scale = 0.62, trim = 0in 0.15in 0in 0.22in, clip=true]
[width=3.5cm]
{/home/jeremic/tex/works/Reports/2015/SEPFEM/figures/Sendai_1Sine_PCk2PC2_elastic_COV45_BaseInput.pdf}
% \caption{Single sine pulse excitation of the soil column at its base.}
% \label{Sendai_1Sine_PCk2PC2_elastic_COV45_BaseInput}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Uncertain Elastic Response at the Surface {\tiny(COV = 120\%)} }
\vspace*{2mm}
\begin{figure}[H]
\centering
\includegraphics
% [scale = 0.62, trim = 0in 0.16in 0in 0.26in, clip=true]
[width=9.3cm]
{/home/jeremic/tex/works/Reports/2015/SEPFEM/figures/Sendai_1Sine_PCk2PC2_elastic_COV120.pdf}
% \caption{Mean $\pm$ std. deviation of the elastic response at the top of the
% soil column for the conservative COV case (COV = 120\%).}
% \label{Sendai_1Sine_PCk2PC2_elastic_COV120}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Displacement PDFs at the Surface {\tiny(COV = 120\%)} }
%\vspace*{20mm}
%\hspace*{30mm}
\begin{figure}[H]
\centering
\includegraphics
%[scale = 0.62, trim = 0in 0.16in 0in 0.26in, clip=true]
[width=10cm]
{/home/jeremic/tex/works/Reports/2015/SEPFEM/figures/Sendai_1Sine_PCk2PC2_elastic_COV120_FullPdf.jpg}
% \caption{Mean $\pm$ std. deviation of the elastic response at the top of the
% soil column for the conservative COV case (COV = 120\%).}
% \label{Sendai_1Sine_PCk2PC2_elastic_COV120}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Displacement CDFs (Fragilities) at the Surface {\tiny(COV = 120\%)} }
\vspace*{3mm}
\begin{figure}[H]
\centering
\includegraphics
% [scale = 0.62, trim = 0in 0.16in 0in 0.26in, clip=true]
[width=11cm]
{/home/jeremic/tex/works/Reports/2015/SEPFEM/figures/Sendai_1Sine_PCk2PC2_elastic_COV120_FullCdf.jpg}
% \caption{Mean $\pm$ std. deviation of the elastic response at the top of the
% soil column for the conservative COV case (COV = 120\%).}
% \label{Sendai_1Sine_PCk2PC2_elastic_COV120}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Probability of Exceedance, $disp=0.1m$ {\tiny(COV = 120\%)} }
\begin{figure}[H]
\centering
\includegraphics
% [scale = 0.62, trim = 0in 0.16in 0in 0.26in, clip=true]
[width=9cm]
{/home/jeremic/tex/works/Reports/2015/SEPFEM/figures/Sendai_1Sine_PCk2PC2_elastic_COV120_ProbExc.pdf}
% \caption{Mean $\pm$ std. deviation of the elastic response at the top of the
% soil column for the conservative COV case (COV = 120\%).}
% \label{Sendai_1Sine_PCk2PC2_elastic_COV120}
\end{figure}
\end{frame}
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary}
\subsection*{Summary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Concluding Remarks}
\begin{itemize}
%\vspace*{2mm}
\item Uncertainty influences results of numerical predictions
\vspace*{2mm}
\item Uncertainty (modeling and parametric) {must}
be taken into account
\vspace*{2mm}
\item Goal is to {predict} and {inform}, not fit
%\vspace*{0.1cm}
\vspace*{2mm}
\item Philosophy of modeling and simulation system \\
{\bf Real ESSI} simulator \\
%
({\small aka: {\cyrssDvanaest Vrlo Prosto},
{ Muy F{\'a}cil},
{Molto Facile},
\raisebox{1.2mm}{\includegraphics[height=4mm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/Real_ESSI_in_different_langauges/Real_ESSI_Chinese.jpeg}},
\raisebox{1.2mm}{\includegraphics[height=4mm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/Real_ESSI_in_different_langauges/Real_ESSI_Japanese.jpg}},
{\greektext{Pragmatik'a E'ukolo}},
\raisebox{1.2mm}{\includegraphics[height=4.5mm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/Real_ESSI_in_different_langauges/Real_ESSI_Hindi.jpg}},
\raisebox{1.2mm}{\includegraphics[height=4mm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/Real_ESSI_in_different_langauges/Real_ESSI_Farsi.jpg}},
{Tr{\`e}s Facile},
{\cyrssDvanaest Vistinski Lesno},
{Wirklich Einfach}
})
% \vspace*{0.4cm}
% \item Collaborators:
% Dr. Budnitz (LBNL),
% Mr. Orbovi{\'c} (CNSC),
% Prof. Pisan{\`o} (TU Delft),
% Prof. Sett (UB),
% Mr. Watanabe (Shimizu)
% and
% UCD students:
% Mr. Abell,
% Mr. Karapiperis,
% Mr. Feng,
% Mr. Sinha,
% Mr. Luo
%
%
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Acknowledgement}
\begin{itemize}
\vspace*{0.1cm}
\item Funding from and collaboration with the USNRC, USDOE, USNSF, CNSC,
AREVA NP GmbH, and Shimizu Corp. is greatly appreciated,
\vspace*{4mm}
\item Collaborators:
% Dr. Budnitz (LBNL),
Prof. Kavvas (UCD),
Prof. Pisan{\`o} (TU Delft),
Mr. Watanabe (Shimizu),
Mr. Vlaski (AREVA NP GmbH),
Mr. Orbovi{\'c} (CNSC)
and
UCD students:
Mr. Abell,
Mr. Karapiperis,
Mr. Feng,
Mr. Sinha,
Mr. Luo
%
%
\end{itemize}
\end{frame}
\end{document}
%
%