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%\maketitle
\title{High fidelity modeling and simulation of SFS
interaction: \\
energy dissipation by design}
\author{\large {Boris Jeremi{\'c}}\\
%\author{\large {Boris Jeremi{\'c} and Nima Tafazzoli}\\
{\em University of California, Davis, CA, 95616, U.S.A.}\\
~\\
published in a book: \\
Soil-Foundation-Structure Interaction, \\
Edited by R.P. Orense, N. Chouw, and M. Pender, \\
CRC Press, Taylor \& Francis Group, 2010.
}
%\author{\large Mahdi Taiebat\\
%{\em University of British Columbia, Vancouver, BC, Canada}\\
%}
\date{}% No date.
\abstract{It is hypothesized that interplay of earthquake, soil, foundation
and structure (SFS) dynamic characteristic, and their interaction in time
domain,
control the behavior of SFS system during earthquakes. Moreover, (passive and
active) control of spatial and temporal location of seismic energy dissipation
(preferably in soil) can improve safety and economy of SFS systems. Such energy
dissipation by design requires high fidelity modeling and simulations. This
paper briefly describes modeling and simulation aspects of energy flow in SFS
system. In addition to that, examples of directed energy dissipation are
presented that show how soil can be used for the benefit of overall SFS system
response to seismic excitation.}
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\section{INTRODUCTION}
Seismic behavior of soil-foundation-structure (SFS) systems has recently gained
increased attentions.
%
Improvements in modeling and simulation technology currently allow
modeling and simulations of a complete SFS interaction with high
fidelity.
%
These models and simulations allow us, in turn, to gain better understanding of
seismic response of the SFS system.
%
Moreover, such high fidelity models and simulations allow us to design the SFS
system(s).
%
Of particular interest is the notion that a designer can/should be able to
direct/design the location, in time and space, where dissipation of seismic energy
takes place.
%
This notion is based on understanding that incoming seismic energy
affects both soils and structures.
%
While focus is usually on structural performance, interaction of soil with
foundation/structure plays a very important (crucial) role in seismic response.
%
The idea is that while energy dissipation in structure (and its components)
leads to damage, and potentially failure, soil medium offers significant energy
dissipation capacity and benefits.
In this paper we briefly discuss seismic energy dissipation
mechanisms in soils and their potential use in improving seismic
soil-foundation-structure system performance.
While SFS interaction has been modeled and simulated for a number of years.
%
We mention a number of references related to SFS interaction importance and
modeling, starting with the very first mention of SFS interaction beneficial
and detrimental effects by Late Prof. Suyehiro \shortcite{Suyehiro1932}.
%
A number of researchers have developed and analyzed SFS systems in last 3
decades, and we mention some of them:
\shortcite{Chen1977},
%\shortcite{Tyapin2007},
\shortcite{Makris1994},
%\shortcite{Sweet1993},
%\shortcite{Dendrou1985},
\shortcite{McCallen1994},
%\shortcite{Zheng1995},
\shortcite{Gazetas98},
%\shortcite{Mylonakis2006},
\shortcite{Mylonakis1997},
%\shortcite{Koo2003},
\shortcite{Fenves1998},
%\shortcite{Shirkhande1999},
%\shortcite{Small2001},
%\shortcite{Tongaonkar2003},
%\shortcite{Chouw2005},
%\shortcite{Kotsoglou2007},
%\shortcite{Soneji2008},
%\shortcite{Stehmeyer2008},
%\shortcite{Saadeghvaziri2008},
\shortcite{Elgamal2008},
%\shortcite{Kappos2002},
\shortcite{Jeremic2003a},
\shortcite{Jeremic2007d},
\shortcite{Jeremic2008a},
%\shortcite{Spyrakos1992},
%\shortcite{Saadeghvaziri2000},
%\shortcite{Ellis2001},
%\shortcite{Mylonakis2006a},
%%\shortcite{Song2006},
%\shortcite{Xia2006},
%\shortcite{Liao2006},
%\shortcite{Sarrazin2005},
%\shortcite{Chai2003},
%\shortcite{Savin2003},
%%`\shortcite{Rassem1997},
%\shortcite{Zhang2002}.
% It is very important to note that assumed beneficial role of not performing
% a full SFSI analysis has been
% turned into dogma. For example the NEHRP-94 seismic code states that:
% %
% {\it ''These [seismic] forces therefore can be evaluated conservatively
% without the adjustments recommended in Sec. 2.5 [i.e. for SSI effects]''}.
% %
In this paper we briefly describe energy dissipation mechanisms for SFS
system.
%
In addition to that, it is claimed that only high fidelity models can be used
for such model based simulation (design) of energy dissipation.
Our main hypothesis is that the interplay of earthquake (nonlinear seismic wave
propagating from source to the structure of interest) with soil and the
structure plays major role in potentially catastrophic failures, but also in
success.
%
Timing and spatial location of energy dissipation within the SFS system, determines
amount of damages and in general controls survivability of structure during
earthquake.
%
If timing and spatial location of energy dissipation can be controlled, one
could optimize the SFS system for safety and economy.
%
This is particularly true if energy dissipation can be directed to soil instead
of foundation and structure.
Directing (by design) energy dissipation for SFS systems requires
development and simulations on high fidelity numerical models.
%
There are a number of cases where interaction of SFS and
dissipation of seismic energy in soil can be deduced by observing structural
damage.
%
The very notion that soil SFS has significant role in dynamic response of
structures comes from Professor Kyoji Suyehiro, (a Naval Engineer turned
Earthquake Engineer,
following his personal experience of Great Kant{\= o} earthquake
(11:58am(7.5), 12:01pm(7.3), 12.03pm(7.2) (shaking until 12:08pm), 1st. Sept.
1923, in Tokyo) who reported $4 \times$ (four) more damage to soft wooden
buildings on soft ground than same building on stiff ground \shortcite{Suyehiro1932}.
%
This was probably due to the close to resonance of building (soft) with
foundation soil (soft) with a long lasting (soft, long period) earthquake.
%
Many years later, \shortcite{Trifunac1998}, show how during Northridge earthquake,
areas with damage to buildings (signifying structural damage) was quite
nicely separated from areas of water pipe breaks (signifying much plasticity and
energy dissipation in soil).
%
In this case, much energy is dissipated in the (soft) soil, never making it to
the building, while for stiff soil do not have such energy dissipation capacity,
transmitting such energy to the building for dissipation (damage).
There are many other cases where such phenomena is observed.
%
Our primary goal here is to emphasize how high fidelity modeling and simulations
of SFS systems can help understand mechanics of such interactions.
%
In addition to that, we use high fidelity models to present
examples of interplay of earthquake, soil and structure dynamic characteristics,
together with the location and timing of energy dissipation.
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\section{SEISMIC ENERGY INPUT AND DISSIPATION}
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\subsection{\it Seismic energy input into SFS system}
Earthquakes release large amounts energy at the source\footnote{
for example, some of the recent large earthquake energy releases are listed:
Northridge, 1994, $M_{Richter} = 6.7$, $E_r = 6.8 \times 10^{16}J$;
Loma Prieta, 1989, $M_{Richter} = 6.9$, $E_r =1.1 \times 10^{17}J$;
Sumatra-Andaman, 2004, $M_{Richter} = 9.3$, $E_r =4.8 \times 10^{20}J$;
Valdivia, Chile, 1960, $M_{Richter} = 9.5$, $E_r =7.5 \times 10^{20}J$;}
%
Part of released energy is radiated as mechanical waves ($\approx 1.6 \times
10^{-5}$) and part of that energy makes it to the surface where SFS system is
located.
Mechanical seismic wave energy enters the SFS system through a closed surface
$\Gamma$ that encompasses (significant) soil volume as well as foundation system
and the structure (Fig.~\ref{DRM}).
%
\begin{figure}[!hbpt]
\vspace*{-7cm}
\begin{center}
\includegraphics[width=9.5cm]{DRMidea03.pdf}
\caption{\label{DRM}\small Geometry of the SFS system. }
\end{center}
\vspace*{-0.5cm}
\end{figure}
%
Kinetic energy flux through closed surface $\Gamma$ includes both incoming
and outgoing waves and can be calculated using Domain Reduction Method
\shortcite{Bielak2001} as:
%
\begin{eqnarray*}
& &E_{flux} =
\nonumber \\
& & \left[0 ; -M^{\Omega+}_{be} \ddot{u}^0_e-K^{\Omega+}_{be}u^0_e ;
M^{\Omega+}_{eb}\ddot{u}^0_b+K^{\Omega+}_{eb}u^0_b \right]_i
\times u_i
\end{eqnarray*}
%
where $M^{\Omega+}_{be}$, $M^{\Omega+}_{eb}$, $K^{\Omega+}_{be}$,
$K^{\Omega+}_{eb}$ are mass and stiffness matrices, respectively for a single
layer of elements just outside of the boundary $\Gamma$, while $\ddot{u}^0_e$
and $u^0_e$ are accelerations and displacements from a free field model for
nodes belonging to that layer of elements.
Alternatively, energy flux can be calculated using \shortcite{Aki2002}:
%
\begin{eqnarray*}
E_{flux} = \rho A c \int_0^t \dot{u}_i^2 dt
\end{eqnarray*}
%
Outgoing kinetic energy can be obtained from outgoing wave field $w_i$, (from
DRM, \shortcite{Bielak2001}), while the difference then represents the incoming
kinetic energy that needs to be dissipated with SFS region.
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\subsection{\it Seismic energy dissipation in SFS system}
Seismic energy that enters the SFS system will be dissipated in a number of
ways.
%
part of the energy that enters SFS system can be reflected back into domain
outside $\Gamma$ by
%
\begin{itemize}
\item wave reflection from impedance boundaries (free surface, soil/rock
layers...).
\item SFS system oscillation radiation.
\end{itemize}
%
While the rest of seismic energy is dissipated through one of the following
mechanisms within SFS domain:
\begin{itemize}
\item Elasto-plasticity of soil
\item Viscous coupling of porous solid with pore fluid (air, water)
\item Elasto-plasticity/damage of the foundation system
\item Elasto-plasticity/damage of the structure
\item viscous coupling of structure with surrounding fluids (air, water)
% \item potential and kinetic energy
% \item potential $\leftarrow \! \! \! \! \! \! \rightarrow$ kinetic energy
\end{itemize}
It is also important to note that in numerical simulations (advocated and used
in this work), part of the energy can be dissipated or produced by purely
numerical means. That is, numerical energy dissipation (damping) or production
(negative damping) has to be carefully controlled \shortcite{local-87},
\shortcite{local-86}.
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\paragraph{Energy Dissipation by Plasticity.}
Elastic-plastic deformation of soil, foundation and structure is probably
responsible for major part of the energy dissipation for large earthquakes.
This, displacement proportional dissipation is a result of dissipation of
plastic work ($W = \int
\sigma_{ij} d \epsilon_{ij}^{pl}$) and is present in all three components of the
system (soil, foundation and the structure). Ideally, majority of the incoming
energy would be dissipated in soil, before reaching foundation and structures.
The possibility to direct energy dissipation to soil can be used in design by
recognizing energy dissipation capacity for different soils. For example,
simple elastic-plastic models of stiff and soft clay as well as dense and
loose send predict different energy dissipation capacities, as shown in
Figure~\ref{EnergyDissipationCapacity}, for single loading-unloading-reloading
cycle.
%
\begin{figure}[!hbpt]
\vspace*{-0.5cm}
\begin{center}
\includegraphics[width=9.5cm]{Energy-Capacity2.pdf}
\vspace*{-0.5cm}
\caption{\label{EnergyDissipationCapacity} \small Energy dissipation capacity
for one cycle at various strains for four generic soils.}
\end{center}
\vspace*{-0.5cm}
\end{figure}
%
While Figure~\ref{EnergyDissipationCapacity} shows that stiff clay and dense
sand have much higher dissipation capacity, it is important to note that soft/loose
soils can undergo much larger deformation/strain, thus offering increased
energy dissipation capacity through flexibility.
%
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\paragraph{Energy Dissipation by Viscous Coupling.}
Viscous coupling of pore fluid (air,
water...) and soil particles and/or foundation or structural components
is responsible for velocity proportional energy dissipation. In particular,
viscous coupling of porous solid and fluid results in $E_{vc}= n^2 k^{-1}
(\dot{U}_i - \dot{u}_i)^2$ energy loss per unit volume.
%
It is noted that this type of dissipation is realistically modeled
using $u-p-U$ formulation \shortcite{Jeremic2007e}.
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\paragraph{Numerical Energy Dissipation and Production.} As noted above,
numerical integration of nonlinear equations of motions affects calculated
energy in various ways. Most common effect for nonlinear (elastic-plastic)
systems is the positive (energy dissipation) and negative (energy production)
damping.
%
For example Newmark (N) \shortcite{Newmark1959} and Hilber--Hughes--Taylor
(HHT) \shortcite{Hilber1977} are energy preserving for linear elastic system with
proper choice of constants ($\alpha=0.0; \beta = 0.25, \gamma = 0.5$).
Both methods can also be used to dissipate higher frequency modes for
linear elastic models by changing constants so that for N: $\gamma \ge
0.5, \;\;\; \beta = 0.25(\gamma+0.5)^2$, while for HHT: $-0.3\dot{3}\le\alpha
\le0, \;\;\;\gamma = 0.5(1-2\alpha), \;\;\; \beta = 0.25(1-\alpha)^2$.
%
However, for nonlinear problems it is impossible to maintain energy of the
system throughout computations \shortcite{local-87}.
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\subsection{UNCERTAINTY ASPECTS}
\label{UA}
Uncertainty of soil material parameters and forcing represents a significant
source of uncertainty of a final computed (simulated) response of SFS
system. Recent development of
Probabilistic Elasto-Plasticity (PEP) and Spectral Stochastic Elastic-Plastic
Finite Element Method (SSEPFEM)
(\shortcite{Jeremic2005a}, \shortcite{Sett2005a}, \shortcite{Sett2005b},
\shortcite{Jeremic2008c}, \shortcite{Sett2009a}, \shortcite{Sett:2009b}) allows
accurate analysis of influence of uncertain soil properties and forcing
on seismic response. Calculation of seismic energy (propagation and dissipation)
is affected by such, ever present uncertainties and such uncertainties should
be taken into
account as best as possible, Above cite (already) published papers and a number of
near future papers (under review) present development of methodology for forward
and backward propagation of uncertainties in dynamic (and static) simulation of
elastic-plastic solids made of (geo-)materials. Such newly
developed, highly accurate, numerical methdology for treatment of material
(left hand side) and forcing (right hand side) uncertainty allows for full
quantification of stochastic (probabilistic) aspects of SFS interaction.
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\section{SELECT EXAMPLES OF ENERGY DISSIPATION}
This section briefly describes two examples of SFS system modeling, simulation
and energy dissipation.
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\paragraph{Use of Soft Soil.}
Simulations on high fidelity model for bridge
SFS system \shortcite{Jeremic2008d} were used to investigate energy flow and
dissipation. Proper modeling of nonlinear wave propagation required large
number of elements and DOFs ($1.6 \times 10^6$ for largest model). Such large
models required development of efficient parallel finite element methodology
(Plastic Domain Decomposition, PDD)
that could handle elastic-plastic computations on multiple generation
distributed memory parallel computers including DataStar at SDSC, Longhorn at
TACC and our own GeoWulf at UCD \shortcite{Jeremic2007d},
\shortcite{Jeremic2008a}.
%
Great care was taken to develop high fidelity model for both soil, foundation
and the structure. Seismic waves were input into the model using DRM
\shortcite{Bielak2001}, and no numerical damping was used, leaving energy
dissipation to elasto-plasticity and radiation damping.
%
Figure~\ref{Mesh} shows a detailed FEM model.
%
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=9.0cm]{PrototypeMesh.jpg}
\caption{\label{Mesh} \small Detailed finite element model of a SFS system.}
\end{center}
\vspace*{-0.5cm}
\end{figure}
%
%
It is important to note that a full (numerical) construction process was
performed, with soil self weight applied first, followed by excavation and
pile installation, pile self weight application, with structure construction
(self wight) application preceding application of seismic input via DRM.
Figure~\ref{Northridge} shows moment response (upper) of the top of bent \#~1,
contrasted with relative velocity energy (lower) for the same bent.
%
%
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=7.5cm,height=3.8cm]{MomentBent1Pile1_25s_SC.pdf}
\\
\includegraphics[width=9.5cm,height=7.5cm]{Northridge-Diff-Comp.pdf}
\vspace*{-4cm}
\caption{\label{Northridge} \small Bending moment response for bent \#~1 (left column)
(top) and relative velocity energy (lower). }
\end{center}
\vspace*{-0.5cm}
\end{figure}
%
Two cases
are analyzed, CCC is a case with all foundations (piles) in a soft clay (Bay mud)
while SSS is for all foundations (piles) resting in dense send soil. Input
motion is from Northridge earthquake, characterized with fairly high energy
input in higher frequencies (stiff earthquake).
%
It is obvious that soft soil dissipates seismic energy by plasticity and that
SFS system in soft clay does not sustain much damage (possibly one case of
plastic yielding on top of bent, at $t$ between 14 and 15 seconds. On the other
hand, in stiff sand, soil does not dissipate much seismic energy, hence bent
\#~1 suffers much plastic yielding (plastic hinge development between $t$ 8
until 12 seconds. It is noted also that the dynamic characteristics of stiff
earthquake, with stiff soil and stiff structure contribute to early close to
resonance response and increase damage. Relative velocity energy plot
(Fig.~\ref{Northridge}, lower) presents similar information, this time in terms
of kinetic energy, that is dissipated through plastic work. Note early peaks for
SSS SFS system, that get dissipated by plastic hinging, while sole peak for CCC
SFS system contributes to one sided plastic hinge development at $t\approx14s$.
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\paragraph{Use of Liquefaction.}
Liquefaction has been consistently put in negative connotation in geotechnical
earthquake engineering. There are many cases where
liquefaction is to be blamed for unacceptable SFS system performance (
\shortcite{Youd89},
%\shortcite{Hamada92},
\shortcite{Yokoyama97},
\shortcite{Berril97},
\shortcite{Kawakami66},
\shortcite{Hamada92a},
\shortcite{Hamada92b},
\shortcite{JSCE66}),
%
However, there is not much evidence (it was not searched for) that liquefaction
actually provided benefit by decreasing (damping out) ground motions. A simple
example is used to illustrate this idea \shortcite{Taiebat2008a}.
%\shortcite{Taiebat2009a}.
%
Figure~\ref{two_soil_columns} presents two models for 1D seismic wave
propagation, namely one (left) with all dense
sand, while the other one (right) is dense sand on top of loose sand layers.
%
%
\begin{figure}[!htbp]
\begin{center}
\vspace*{-0.5cm}
\includegraphics[width=11cm,angle=90]{Mesh-Isolation.pdf}
%\\
\caption{\label{two_soil_columns} \small Two soil column models, left is all dense
sand, right is dense sand on top of loose sand layers. Seismic motions applied
to the bottom are also shown.}
\end{center}
\vspace*{-0.5cm}
\end{figure}
%
%
Seismic wave is propagated through the soil (input is also shown in
Fig.~\ref{two_soil_columns}) with resulting acceleration records at different
soil depths shown in Figure~\ref{two_soil_columns_results}.
%
\begin{figure}[!htb]
%\vspace*{-3cm}
\begin{center}
\includegraphics[width=9.5cm]{time-history-acc.jpg} \\
\caption{\label{two_soil_columns_results} \small Acceleration time history, at
different soil levels. Left is all dense sand model, right is dense with loose
bottom sand layer.}
\end{center}
\vspace*{-0.5cm}
\end{figure}
%\hspace*{2.3cm} EPPR \hspace{1.5cm} $\gamma$ \hspace{1.7cm} $u_{hor}$
Since bottom loose soil layers do liquefy (from effective stress results),
seismic energy does not propagate much above bottom layers. Main dissipation
mechanisms are related to soil plasticity and coupling of solid skeleton with
pore fluid.
%
Figure~\ref{two_soil_columns_energy} shows measured (simulated) kinetic energy
at the top of both soil models. Layered model (with loose, liquefiable
layer at the bottom) has reduction of top of model kinetic energy of at least
three times, which might significantly contribute to damage reduction of any
foundation and structure placed on top of such soil system.
%
\begin{figure}[!htb]
\begin{center}
%\vspace*{-1.3cm}
\includegraphics[width=10cm]{StackElements-Compare.pdf}
\vspace*{-4.0cm}
\caption{\label{two_soil_columns_energy} \small Kinetic energy at the top of soil
layers.}
\end{center}
\vspace*{-0.5cm}
\end{figure}
%\hspace*{2.3cm} EPP
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%
%
% \section{UNCERTAINTY ASPECTS}
% \label{UA}
%
%
% Uncertainty of soil parameters represents a significant source of uncertainty of
% a final computed (simulated) response of SFS system. Recent development of
% Probabilistic Elasto-Plasticity (PEP) and Spectral Stochastic Elastic-Plastic
% Finite Element Method (SSEPFEM)
% (\shortcite{Jeremic2005a}, \shortcite{Sett2005a}, \shortcite{Sett2005b},
% \shortcite{Jeremic2008c}, \shortcite{Sett2009a}, \shortcite{Sett:2009b}) allows
% accurate analysis of influence of soil uncertainty properties on seismic
% response. Calculation of seismic energy (propagation and dissipation) will be
% affected by such, ever present uncertainties as well and should be taken into
% account as best as possible, Above cite (already) published papers and a number of
% near future papers (under review) present development of methodology for forward
% and backward propagation of uncertainties in dynamic (and static) simulation of
% elastic-plastic solids made of (geo-)materials. Such nwely
% developed, highly accurate treatment of material (left hand side) and forcing
% (right hand side) uncertainty allows for full quantification of stochastic
% (probabilistic) aspects of SFS interaction.
%
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\section{SIMULATION PLATFORM}
\label{SP}
Numerical simulations described in this paper were done using sequential
and parallel application programs developed at UCD, with use of a number of
publicly available numerical libraries.
%
Parallel simulation were based on recently developed Plastic Domain
Decomposition (PDD) method \shortcite{Jeremic2007d,Jeremic2008a}.
%
Graph partitioning used in PDD is based on ParMETIS libraries \shortcite{Karypis98c}).
%
Small part of OpenSees framework \shortcite{McKenna97} was used to
connect
the finite element domain. In particular, Finite Element Model Classes from
OpenSees (namely, class abstractions Node, Element, Constraint, Load, Domain and
set of Analysis classes)
where used to describe finite element model and to store results of
analysis
performed on a model. The domain and analysis classes
were significantly modified to improve parallel performance and were
organized as
Modified OpenSees Services (MOSS) library. In addition to that, build
process and organization of libraries was completely redone in order to remove
known limitations of OpenSees program.
%
% An excellent adoption of Actor model
% \citep{Hewitt73,Agha84} and addition of a Shadow, Chanel, MovableObject,
% ObjectBroker, MachineBroker classes within OpenSees framework
% \citep{McKenna97} also provided an excellent basis for our development.
% %
%
On a lower level, a set of Template3Dep numerical libraries
\shortcite{Jeremic2000f}
were
used for constitutive level integrations, nDarray numerical libraries
\shortcite{Jeremic97d} were used to handle vector, matrix
and tensor manipulations, while FEMtools element libraries from UCD
CompGeoMech toolset \shortcite{Jeremic2004d} were used to supply other
necessary libraries and components.
%
%
%
Parallel solution of system of equations has been provided by PETSc set of
numerical libraries \shortcite{petsc-web-page,petsc-user-ref,petsc-efficient}).
%
Application programs used for simulation were created by linking above
mentioned libraries in the Finite Element Interpreter (\FEI{}).
%
Large part of simulation was carried out on our local sequential computers
and and our parallel
computer GeoWulf.
Only the largest models (too big to fit on GeoWulf system) were simulated
on TeraGrid machine at SDSC and TACC.
%
%It should be noted that the software
%part of the simulation platform (executable program) is available through
%Author's web site.
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\section{CONCLUSIONS}
Interplay of Earthquake, Soil, Foundation and Structure dynamics in time domain
plays a major role in catastrophic failures and great successes.
%
High fidelity modeling and simulation offers an unprecedented opportunity
to improve design.
%
The ability to model and simulate flow of seismic energy in the SFS
system with high fidelity, makes it possible to design energy dissipation
in most economical way, in soil,
%
Directing, in space and time, seismic energy flow in the SFS system will
lead to increase in safety and economy.
%
The main purpose of this brief paper was to overview modeling and simulations
issues and show illustrative examples of directing energy flow for SFS systems.
It is hoped that public domain modeling and simulations tools, such as
\FEI{} and recently developed \url{www.OpenHazards.com} will be used more in
future to increase
safety and reduce cost of infrastructure objects in earthquake prone areas.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{ACKNOWLEDGMENT}
%
I would like to thank Professors
Zhaohui Yang (University of Alaska),
Mahdi Taiebat (University of British Columbia),
Kallol Sett (University of Akron)
and Drs.
Zhao Cheng (EarthMechanics Inc.),
Guanzhou Jie (Wells Fargo Securities),
Matthias Preisig (Ecole Polytechnique F{\'e}d{\'e}rale de Lausanne)
and graduate student researcher
Mr. Nima Tafazzoli (UCD)
for their contributions to this paper.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\newpage
\renewcommand{\baselinestretch}{0.9}
\small % trick from Kopka and Daly p47.
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