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\vspace*{0.0truecm}
\begin{center}
{\huge \bf
%\vspace*{1.5truecm} \\
\rule[3mm]{0cm}{1.3cm} The Plastic Domain Decomposition \\
}
\end{center}
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\begin{large}
\begin{center}
{\bf Boris Jeremi{\' c}}
\end{center}
\end{large}
\vspace*{0.50truecm}
\begin{small}
\begin{center}
{Department of Civil and Environmental Engineering }\\
{University of California, Davis}\\
\end{center}
\end{small}
%
%\vspace*{0.50truecm}
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\begin{small}
\begin{center}
{MRCCS/NSF Summer School} \\
{High Performance Computing in Finite Element Analysis}\\
{1st  5th September 2003,}\\
{University of Manchester}\\
\end{center}
\end{small}
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\begin{small}
\begin{description}
%\item Supported in part by the NSFPEER \#~EEC9701568,
%CalEPA \#~9901337V,
\item Supported in part by the NSF, PEER,
Caltrans, and CalEPA.
\item Collaborators:
Professors
Zhaohui Yang (UAA),
Sashi Kunnath (UCD),
%Bruce Kutter (UCD),
Gregory Fenves (UCB),
Jacobo Bielak (CMU),
%Bernd Hamann (UCD),
Zhaojun Bai (UCD),
George Karypis (UMN),
Drs.
Francis McKenna (UCB),
Ingrid Hotz (UCD),
and
graduate students
Xiaoyan Wu (UCD)
Ritu Jain (UCD),
Zhao Cheng (UCD),
Kallol Sett (UCD),
Qing Liu (UCD),
Jinxiu Liao (UCD).
Guanzhou Jie (UCD).
\end{description}
\end{small}
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\slide{Motivation}
\begin{large}
\begin{itemize}
\vspace*{1.0cm}
\item Create high fidelity models of constructed facilities
(bridges, buildings, port structures, dams...).
\vspace*{1.0cm}
\item Models will live concurrently with the physical system
they represent.
\vspace*{1.0cm}
\item Models to provide owners and operators with the
capabilities to assess operations and future performance.
\vspace*{1.0cm}
\item Use observed performance to update and validate models
through simulations.
\end{itemize}
\end{large}
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\slide{Large Scale Numerical Simulation Goal}
\begin{large}
\begin{itemize}
\vspace*{0.5cm}
\item Scalable parallel finite element method for inelastic computations (solid
and structural elements)
\vspace*{0.5cm}
\item Based on state of the art computational mechanics theories and implementation
\vspace*{0.5cm}
\item Available for a range of sequential and parallel machines, including
clusters, grids of machines and clusters (DMPs), and also multiprocessors (SMPs)
\vspace*{0.5cm}
\item Public domain, portable platform.
\end{itemize}
\end{large}
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\slide{Incomplete Historical Background}
%\vspace*{0.3cm}
\begin{large}
\begin{itemize}
\item Substructuring to achieve partitions (Noor et al. \cite{Noor1978}, Utku et
al. \cite{Utku1982} and Storaasil and Bergan \cite{Storaasli1987})
%
\item Techniques to account for
different types of elements but used substructures of same element types
(nonbalanced computations) (Fulton and Su \cite{Fulton1992} ).
%
\item dynamic analysis of framed structures with the objective of minimizing
communications (Hajjar and Abel \cite{Hajjar1988})
\item
parallel computational techniques for
elasticplastic problems but tied the algorithm to the specific multiprocessor
computers used (and specific network connectivity architecture) (Klaas et al. \cite{Klaas1994})
\item Greedy domain partitioning algorithm good for topological DD but does not
redistributed the domains as a function of developed nonlinearities (Farhat \cite{Farhat1987})
\end{itemize}
\end{large}
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\slide{Brief Methodology Background}
%\vspace*{0.3cm}
\begin{large}
\begin{itemize}
\item Static domain decomposition will lead to optimal parallel run for
many problems
\begin{itemize}
\item equal number of mesh elements to each processor, this will balance
computational load (CL) on parallel machine,
\item minimize the size of subdomain boundaries, this will minimize the
interprocessor communications overhead,
\end{itemize}
\item This can be done many ways, recently a set of good graphs partitioner were
developed (METIS family by Karypis et al.)
\item This is indeed good for problems where the computational domain or
discretization does not change.
\item For inelastic computations using finite element method, the change will
occur in internal state determination.
\end{itemize}
\end{large}
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\slide{Inelastic Parallel Problem}
\begin{large}
\begin{itemize}
%\vspace*{0.2truecm}
\item Presence of elastic and inelastic (plasticity, damage) computational domains.
%\vspace*{0.2truecm}
\item Difference in computational load for elastic and inelastic state
determination leads to computational load imbalance
%\vspace*{0.2truecm}
\item This leads to imbalanced computations, very inefficient, not much gain
from using parallelization
%\vspace*{0.2truecm}
\item Internal state determination can take as much as 80\% of CL
\end{itemize}
%
\begin{figure}[!htb]
%\vspace*{1.1cm}
\begin{center}
{\includegraphics[width=3truecm]{/home/jeremic/tex/works/Conferences/2003/ParallelManchester/sii.pdf}}
\end{center}
\vspace*{0.3cm}
\end{figure}
\noindent
\end{large}
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\slide{Inelastic Computational Load Balancing Challenge:
\\
Adaptive Computations}
\begin{large}
\begin{itemize}
\item Dynamic computations, structure of elastic and elasticplastic
domains changes dynamically and unpredictably
\item Periodic computational
loadbalancing is required during the course of the computation
\item Computational load balancing cost similar to static DD (balance the
mesh elements and minimize the interprocessor communications
\item It also requires that the
cost associated with redistributing the data in order to balance
the computational load is minimized
\end{itemize}
\end{large}
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\slide{Inelastic Computational Load Balancing Challenge:
\\
Multiphase Computation }
\begin{large}
\begin{itemize}
\vspace*{0.7truecm}
\item Elasticplastic computations follow up the
elastic computations, there is a synchronization step between
\vspace*{0.7truecm}
\item Each phase needs to be separately computational load balanced
\begin{itemize}
\item Elastic phase computations (topological DD might be OK)
\item Elasticplastic phase computations (PDD)
\end{itemize}
\vspace*{0.7truecm}
\item System of equations needs to be CL balanced as well
\end{itemize}
\end{large}
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\slide{Elastic Computations}
\begin{large}
%
\begin{samepage}
\begin{eqnarray*}
{}^{(m)}\! K_{IacJ}
=
\int_{V^{m} } H_{I,b} \; E_{abcd} \; H_{J,d} \; dV^{m}
\label{FED20}
\end{eqnarray*}
\end{samepage}
%
%
%
\begin{samepage}
\begin{eqnarray*}
E_{ijkl} = \lambda \delta_{ij} \delta_{kl}
+ \mu \left( \delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk} \right)
\label{elast06}
\end{eqnarray*}
\end{samepage}
%
%
\noindent
{\em Lam{\'e} (elastic) coefficients}:
%
%
\begin{samepage}
\begin{eqnarray*}
\lambda = \frac{\nu E}{\left( 1+\nu \right) \left( 12\nu \right)}
\;\;\;\;\;
;
\;\;\;\;\;
\mu = \frac{E}{ 2 \left( 1+\nu \right) }
\label{elast07}
\end{eqnarray*}
\end{samepage}
%
%
%
Applies to linear and nonlinear elasticity
\end{large}
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\slide{ElasticPlastic Computations}
\begin{large}
\begin{itemize}
\vspace*{0.4truecm}
\item Integration of constitutive equations at each integration point:
\begin{itemize}
\item Explicit integration (Forward Euler (FE) algorithm)
\item Implicit integration (Backward Euler (BE) algorithm)
\item Midpoint integration (CrankNickolson (CN) algorithm)
\end{itemize}
\vspace*{0.4truecm}
\item Single step (FE) and iterative algorithms (BE, CN)
\vspace*{0.4truecm}
\item Elastic computations, computational load per element (integration
point) known a priori
\vspace*{0.4truecm}
\item ElasticPlastic computations, computational load much larger
\vspace*{0.4truecm}
\item Computations Load is not known a priori (the extent of plastic zone is not
known prior to analysis)
\end{itemize}
%
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\end{large}
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\slide{Implicit Constitutive Integration}
\begin{large}
Iterative procedure, unknown number of steps
%
\begin{eqnarray*}
& & \Delta \sigma_{mn} =
 (
{}^{old} \! r_{ij} +
\\
& &
\frac { {}^{n+1} \! F^{old}

{}^{n+1} \! n_{mn} \; {}^{old} \! r_{ij} {}^{n+1}\!T_{ijmn}^{1} }
{ {}^{n+1} \! n_{mn} E_{ijkl} \; {}^{n+1} \!H_{kl} {}^{n+1}\!T_{ijmn}^{1}

{}^{n+1} \! \xi_{\ast} \; h_{\ast} }
\; E_{ijkl} \; {}^{n+1} \!H_{kl}
)
{}^{n+1}\!T_{ijmn}^{1}
\end{eqnarray*}
%
\begin{eqnarray*}
\Delta q_{\ast} =
\left(
\frac { {}^{n+1} \! F^{old}

{}^{n+1} \! n_{mn} \; {}^{old} \! r_{ij} {}^{n+1}\!T_{ijmn}^{1} }
{ {}^{n+1} \! n_{mn} E_{ijkl} \; {}^{n+1} \!H_{kl} {}^{n+1}\!T_{ijmn}^{1}
 {}^{n+1} \! \xi_{\ast} \; h_{\ast} }
\right)
h_{\ast}
\end{eqnarray*}
%
%
%
\begin{small}
%
\begin{samepage}
\begin{eqnarray*}
{}^{n+1} \!T_{ijmn} =
\delta_{im} \delta_{nj} +
\lambda \; E_{ijkl} \;
\left. % just because of the \right
\frac{ \partial m_{kl} }{\partial \sigma_{mn}} \right_{n+1}
\;\; \mbox{;} \;\;
{}^{n+1} \!H_{kl} =
{}^{n+1} \!m_{kl}
+
\left. % just because of the =right
\lambda \frac{ \partial m_{kl} }{\partial q_{\ast}} \right_{n+1} \! h_{\ast}
\\
n_{ij} = \frac{\partial F}{\partial \sigma_{ij}}
\;\; \mbox{;} \;\;
m_{ij} = \frac{\partial Q}{\partial \sigma_{ij}}
\;\; \mbox{;} \;\;
\xi_{\ast} = \frac{\partial F}{\partial q_{\ast}}
\end{eqnarray*}
\end{samepage}
\end{small}
\end{large}
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\slide{Computational Load Varies}
\begin{large}
\begin{itemize}
\item Load per integration point for each iteration/increment (4 CPU cycles
(CPUc) for addition, 6 CPUc for multiplication, 13 CPUc for division...):
\begin{itemize}
\item Elastic computation: approx. $3600$ CPUc
\item Forward Euler single step: approx. $7000$ CPUc
\item Backward Euler single step: approx. $12000 + n \times 15000 $ CPUc
\end{itemize}
\item Example \#1: say 5 iteration steps $\rightarrow$ $87000 $ CPUc (24 times more CPU load)
\item Example \#2: say 25 iteration steps $\rightarrow$ $387000 $ CPUc (108 times more CPU load)
\item This is all load per one integration point!
\item Standard 20 node brick element should have at least 27 integration points ($3 \times 3 \times 3)$
\end{itemize}
\end{large}
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\slide{Computational Load Imbalance}
%
\begin{large}
\begin{itemize}
%\vspace{0.5cm}
%\item Domain decomposition (DD) technique (substructuring)
\vspace{0.5cm}
\item Existing parallel computational algorithms: topological(preprocessing) DD
into substructures with equal number of elements
\vspace{0.5cm}
\item Topological DD assumes equal computational load per element
\vspace{0.5cm}
\item Topological DD performed during a preprocessing phase
\vspace{0.5cm}
\item No provisions for variation of computational load per element or for
variation in connectivity speed (latency and bandwidth)
\vspace{0.5cm}
\item Separate problem for internal state determination (elasticplastic
computations on the element level) and for the system of equations solution
\end{itemize}
\end{large}
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\slide{Possible Solutions}
%
\begin{large}
\begin{itemize}
%\item Domain decomposition (DD) technique (substructuring)
%\vspace{0.2cm}
\item Perform preprocessing stage DD, topological DD (
equal computational load per element, preprocessing
phase, no provisions for: variation of computational load per element,
connectivity speed, latency and bandwidth)
%\vspace{0.2cm}
\item Distribute small subdomains to processor as they become idle, feed hungry
CPUs (dynamic CL by design, expensive as it sends a lot of "small" data packets,
internal state determination and system solution phases are totally separate,
a lot of data moving around)
\item Element by element family of methods in conjunction with iterative
equation solvers
%\vspace{0.2cm}
\item Dynamic CL balancing $\rightarrow$ Plastic Domain Decomposition (currently
being implemented into OpenSees)
\end{itemize}
\end{large}
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\slide{Feed Hungry CPUs}
%
\begin{large}
\begin{itemize}
\item Distribute computations of element matrices to parallel nodes in small groups
\item Solve the system in parallel using MP\_SOLVE
\item Both phases separate
\end{itemize}
%\vspace{0.5cm}
%\item Relatively slow network hinders performance for 2 nodes
%
\begin{figure}[!htb]
\vspace*{1.1cm}
\begin{center}
{\includegraphics[width=10.2cm]{/home/jeremic/tex/works/Conferences/2000/ASCEEM_Austin/SSI/Present/plab_pdf.jpg}}
\hfill
{\includegraphics[width=10.2cm]{/home/jeremic/tex/works/Conferences/2000/ASCEEM_Austin/SSI/Present/plabMPS_pdf.jpg}}
\end{center}
\vspace*{0.3cm}
\end{figure}
\noindent
stiffness matrix formations \hspace*{4.5cm} system solution
\end{large}
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\slide{Development of the Plastic Domain Decomposition}
%
\begin{large}
\begin{itemize}
\item Based on work of Karypis et al.
\item Multilevel Graph Partitioning
\begin{itemize}
\item graph coarsening
\item initial partitioning
\item multilevel refinement
\end{itemize}
\item ParMETIS system
\item Weighted graphs edges and \\ graph nodes
\item Zoltan framework used as an \\
interface layer for extensibility
\end{itemize}
\begin{figure}[!htb]
\vspace*{18.1cm}
\hspace*{12cm}
{\includegraphics[width=13.2cm]{/home/jeremic/tex/works/Proposals/2002/Parallel01/gp.pdf}}
\vspace*{0.3cm}
\end{figure}
\end{large}
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%
% \slide{Adaptive Graph Partitioning}
%
% %
% \begin{large}
%
% \begin{itemize}
%
% \item Dynamic computational loadbalancing algorithms, based on the
% multilevel graph partitioning paradigm
%
% \item Attempt to minimize the data redistribution costs
%
% \begin{itemize}
% \item compute a new partitioning from scratch and then intelligently map
% this back to the original partitioning (tends to result in partitionings
% that do very well at minimizing the interprocessor communications costs,
% because when a new partitioning is computed from scratch, we can
% use a stateoftheart partitioning method to do so, however it results in
% higher data redistribution costs compared to other methods,
% especially when the partitioning is only slightly imbalanced. This is because it
% is often the case that the newly computed partitioning is substantially
% different from the original partitioning, so even a good remapping of
% the new partitioning can still incur large data redistribution costs)
%
% \item Perturb the original partitioning just enough so as
% to balance the computational load (extremely low data
% redistribution costs, however result in higher interprocessor
% communications costs for highly imbalanced
% partitionings, because during the process of balancing the partitioning,
% locally optimum selections are made and the more that partitioning needs to be
% perturbed in order to balance it, the greater the likelihood that the effect of
% the locally optimum decisions that are made to balance the partitioning will be
% globally suboptimal)
%
% \end{itemize}
%
%
%
% \end{itemize}
%
%
% \end{large}
%
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\slide{Adaptive Graph Partitioning}
%
\begin{large}
\begin{itemize}
\vspace*{1.0cm}
\item Dynamic computational loadbalancing algorithms, based on the
multilevel graph partitioning paradigm
\vspace*{1.0cm}
\item Attempt to minimize the data redistribution costs using one of
the two methods:
\begin{itemize}
\item compute a new partitioning from scratch and then intelligently map
this back to the original partitioning
\item Perturb the original partitioning just enough so as
to balance the computational load
\end{itemize}
\end{itemize}
\end{large}
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\slide{Adaptive Graph Partitioning I}
%
\begin{large}
Compute a new partitioning from scratch and then intelligently map
this back to the original partitioning:
\begin{itemize}
\item result in partitioning
that do very well at minimizing the interprocessor communications costs,
\item because when a new partitioning is computed from scratch, we can
use a stateoftheart partitioning method to do so,
\item results in
higher data redistribution costs compared to other methods,
\item especially when the partitioning is only slightly imbalanced
\item because it
is often the case that the newly computed partitioning is substantially
different from the original partitioning
\item so even a good remapping of
the new partitioning can still incur large data redistribution costs
\end{itemize}
\end{large}
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\slide{Adaptive Graph Partitioning II}
\begin{large}
Perturb the original partitioning just enough so as
to balance the computational load:
\begin{itemize}
\item extremely low data
redistribution costs,
\item result in higher interprocessor
communications costs for highly imbalanced
partitioning,
\item because during the process of balancing the partitioning, locally optimum
selections are made and the more that partitioning needs to be perturbed in
order to balance it, the greater the likelihood that the effect of the locally
optimum decisions that are made to balance the partitioning will be globally
suboptimal
% \item
\end{itemize}
\end{large}
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\slide{Multiphase Graph Partitioning}
\begin{large}
\begin{itemize}
\item Traditional graph partitioners CL balance only single phase
\item Other phases might be seriously imbalanced
\item Generalized formulation of the graph partitioning problem that is able to
balance multiple phases simultaneously, while also minimizing the
interprocessor communications costs
\item Think of adaptive partitioning as a multiobjective optimization
problem, that is minimize both the interprocessor communications, the data
redistribution costs and create good partitions (CL balanced with minimal
boundary)
\item Unified adaptive partitioningrepartitioning algorithms
\item Compute decompositions that take into account the relative costs of
performing interprocessor communications and data redistributions
\end{itemize}
\end{large}
\vspace*{3.5cm}
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\slide{Plastic Domain Decomposition}
\begin{large}
\begin{itemize}
\vspace*{0.5cm}
\item Take into the account:
\begin{itemize}
\item heterogeneous element loads that change in each iteration
\item heterogeneous processor performance (multiple generations nodes)
\item interprocessor communications
\item data redistribution costs
\end{itemize}
\vspace*{0.5cm}
\item Perform global optimization for both internal state determination
and system solution phases
\vspace*{0.5cm}
\item Available for all elements (solid, structural) that provide the
interface (sendSelf, RecvSelf, timer or CL weight estimate)
\vspace*{0.5cm}
\item Able to handle SMPs, local clusters, grids of computers
% \item
\end{itemize}
\end{large}
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\slide{PDD Implementation}
\begin{large}
\begin{itemize}
\item Internal state determination:
\begin{itemize}
\item Use lightweight measurement of each element performance (or provided
information on CL)
\item Use lightweight measurement of network latency and bandwidth
(including local are and wide area networks)
\item Use information from previous incremental step to make PDD decisions
\item Question: do we perform PDD for each iteration (or after few
iterations) or for each increment (or after few increments)
\end{itemize}
\item System of equations solution (SES):
\begin{itemize}
\item Use previous decomposition as much as possible
\item might be somewhat imbalanced but it is OK if SES takes only 20\% of time
\end{itemize}
% \item
\end{itemize}
\end{large}
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\slide{PDD Goals: SFSI}
\begin{figure}[!h]
\vspace*{0.6cm}
\begin{center}
{\includegraphics[width=16cm]{/home/jeremic/tex/works/Thesis/JinxiuLiao/soilmeshDEEP02.jpg}}
\hspace*{7cm}
\\
\vspace*{4cm}
{\includegraphics[width=11cm]{/home/jeremic/tex/works/Presentation/2003/PEERSiteVisit/I880_system.jpg}}
\hfill
{\includegraphics[width=8cm]{/home/jeremic/tex/works/Presentation/2002/SFBayComputingDay/I880_02.jpg}}
\vspace*{4cm}
\end{center}
\end{figure}
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\slide{PDD Working Example}
\begin{large}
Pile, column and a mass on top
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\begin{figure}[!htb]
\vspace*{2.0cm}
\hspace*{14.0cm}
{\includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2003/7USNCCM/PlasticBowl/model3.jpg}}
\end{figure}
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\begin{figure}[!h]
\begin{center}
\vspace*{5.0cm}
{\includegraphics[width=11.5cm]{/home/jeremic/tex/works/Thesis/JinxiuLiao/newthesis/figures/Eq_P/P_T_DL_NS.pdf}}
\hspace{1.8cm}
{\includegraphics[width=11.5cm]{/home/jeremic/tex/works/Thesis/JinxiuLiao/newthesis/figures/Eq_P/P_S_DL_NS.pdf}}
\vspace*{4.2cm}
\end{center}
\end{figure}
\hspace*{4cm} Stiff soil \hspace*{8cm} Soft soil
\end{large}
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\slide{PDD: Elastic Decomposition}
\begin{large}
\begin{itemize}
\item 16 subdomains,
\item approx. equal number of elastic elements per domain,
\item minimized interdomain boundary
\item OpenSees $\Rightarrow$ Zoltan $\Rightarrow$ ParMETIS
% \item
\end{itemize}
\end{large}
%\vspace*{3.5cm}
%\vspace*{3.5cm}
\begin{figure}[htb]
%\vspace*{15.5cm}
\begin{center}
{\includegraphics[height=10.0cm]{/home/jeremic/tex/works/Thesis/RituJain/noweight.jpg}}
%{\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Thesis/RituJain/10weight.jpg}}
%{\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Thesis/RituJain/1000weight.jpg}}
\end{center}
\vspace{9.5cm}
%\vspace*{0.5cm}
\end{figure}
%
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\slide{PDD: Mild Plasticity}
\begin{large}
\begin{itemize}
\item Small amount of plastic elements close to the pile
\item Note smaller domain close to the pile
\item Note also somewhat increased interdomain boundary
\end{itemize}
\end{large}
%\vspace*{3.5cm}
%\vspace*{3.5cm}
\begin{figure}[htb]
%\vspace*{15.5cm}
\begin{center}
%{\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Thesis/RituJain/noweight.jpg}}
{\includegraphics[height=10.0cm]{/home/jeremic/tex/works/Thesis/RituJain/10weight.jpg}}
%{\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Thesis/RituJain/1000weight.jpg}}
\end{center}
\vspace{9.5cm}
%\vspace*{0.5cm}
\end{figure}
%
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\slide{PDD: Severe Plasticity}
\begin{large}
\begin{itemize}
\item Note small number of elements per CL heavy domains
\item Note also two large, mostly elastic, subdomains
\item Good DD for internal state determination, but bad for SES
\item Use graphpartitioner to optimize SES phase using info on imbalance and network speed
% \item
\end{itemize}
\end{large}
%\vspace*{3.5cm}
%\vspace*{3.5cm}
\begin{figure}[htb]
\vspace*{2.0cm}
\begin{center}
%{\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Thesis/RituJain/noweight.jpg}}
%{\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Thesis/RituJain/10weight.jpg}}
{\includegraphics[height=10.0cm]{/home/jeremic/tex/works/Thesis/RituJain/1000weight.jpg}}
\end{center}
\vspace{8.9cm}
%\vspace*{0.5cm}
\end{figure}
%
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\slide{Current Grid Resources}
\begin{figure}[!htbp]
\begin{center}
{\includegraphics[width=21truecm]{/home/jeremic/tex/works/Conferences/2003/ParallelManchester/Grid05.pdf}}
\end{center}
\vspace*{1cm}
%\vspace*{0.3cm}
\end{figure}
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\slide{Stress Field Visualization}
%
\begin{figure}[!htb]
\begin{center}
{\includegraphics[height=8.0truecm]{/home/jeremic/tex/works/Papers/2001/Visualization/pic15.pdf}}
{\includegraphics[height=8.0truecm]{/home/jeremic/tex/works/Papers/2001/Visualization/pic27.pdf}}
\\
\vspace*{0.2cm}
{\includegraphics[height=8.0truecm]{/home/jeremic/tex/works/Papers/2001/Visualization/pic28.pdf}}
{\includegraphics[height=8.0truecm]{/home/jeremic/tex/works/Papers/2001/Visualization/pic38.pdf}}
\vspace*{1cm}
\end{center}
\end{figure}
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\slide{Pile Group Visualization}
%
\begin{figure}[!htb]
\begin{center}
{\includegraphics[height=6truecm]{/home/jeremic/tex/works/Papers/2001/Visualization/pic71.pdf}}
{\includegraphics[height=6truecm]{/home/jeremic/tex/works/Papers/2001/Visualization/pic70.pdf}}
{\includegraphics[height=6truecm]{/home/jeremic/tex/works/Papers/2001/Visualization/pic62.pdf}}
\\
\vspace*{0.2cm}
{\includegraphics[height=9truecm]{/home/jeremic/tex/works/Papers/2001/Visualization/pic33.pdf}}
{\includegraphics[height=9truecm]{/home/jeremic/tex/works/Papers/2001/Visualization/pic80.pdf}}
\end{center}
\vspace*{1cm}
\end{figure}
%
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\slide{Constitutive Integration \\ Visual
Debugging}
%
\begin{figure}[!htb]
\begin{center}
{\includegraphics[width=10cm]{/home/jeremic/oofep/mathematica/BModel/BEBtest.pdf}}
{\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/1999/LineSearch/Erratic02.pdf}}
\vspace*{1cm}
\end{center}
\end{figure}
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\slide{Conclusions}
\begin{large}
\begin{itemize}
\vspace*{1.0cm}
\item Plastic Domain Decomposition for tightly and loosely connected processors
\vspace*{1.0cm}
\item Can handle both computational load imbalance, heterogeneous processors and
heterogeneous connectivity speeds
\vspace*{1.0cm}
\item Use of ParMETIS, Zoltan (and all that interfaces with Zoltan), SuperLU, MP\_SOLVE
\vspace*{1.0cm}
\item Public domain within OpenSees framework
%\vspace*{1.0cm}
%\item
\end{itemize}
\end{large}
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% \includegraphics[height=5.5truecm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/PileGroups/Plastified_3X3pg_skewCOLOR.jpg}
% \includegraphics[height=5.5truecm]{/home/jeremic/public_html/GeoWulf/GeoWulf_01_Nov2001a.jpg}
% \includegraphics[height=5.5truecm]{/home/jeremic/tex/works/Papers/2001/Visualization/pic30.jpg}
% \includegraphics[height=5.5truecm]{/home/jeremic/tex/works/Papers/2001/Visualization/pic71.jpg}
% \\
% \includegraphics[height=6.0truecm]{/home/jeremic/tex/works/Papers/2001/Visualization/pic80.jpg}
% \includegraphics[height=6.0truecm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/MQP_sand_wsfclay_UFmid2hdp0.pdf}
% \includegraphics[height=6.0truecm]{/home/jeremic/tex/works/Reports/2001/PEER/FinalReport/sp2_stress_pile_isoSmall.jpg}
% % \includegraphics[height=5.0truecm]{/home/jeremic/tex/works/Reports/2001/PEER/FinalReport/sp2_MQP_pile.jpg}
% \\
% \includegraphics[height=6.0truecm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/Cube/Cubic_el_05.jpg}
% \includegraphics[height=6.0truecm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/Cube/Cubic_pl_05.jpg}
% \includegraphics[height=6.0truecm]{/home/jeremic/tex/works/Conferences/2002/WCCM/Keynote/Present/pversion_slope.jpg}
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\bye