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\title{Convex Modeling - A Generalization Of Interval Analysis For
Nonprobabilistic Treatment Of Uncertainty}
\author{Isaac Elishakoff}

\maketitle

\begin{abstract} 
This paper reviews some new developments is convex modeling of
uncertainty in structural approaches to engineering. Convex modeling, on
one hand, represents an alternative and complement to traditional
probabilistic methods, and on the other hand, it is a generalization of
the mathematical theory of interval analysis.
\end{abstract}

\auffil{The author is with the Center for Applied Stochastics Research,
Department of Mechanical Engineering, Florida Atlantic University, Boca
Raton, FL 33431-0991. This study has been supported by the National
Science Foundation under award No. MSS-9215698 (Dr. R. C. Chong --
Program Director). Any opinions, findings, and conclusions or
recommendations expressed in this publication are those of the author
and do not necessarily reflect the views of the National Science 
Foundation.}

\section{Probabilistic Models}

A structure can be modeled as a system subject to influences,
disturbances, excitations, and so forth, designated as {\it inputs}. We
are usually interested in the response of the system: its dynamic
displacements, stresses, or strains designated as {\it outputs.} In
formulating the probabilistic model, we first identify the random
variables and/or functions, which can be the characteristics of the
inputs and/or of the system itself. To perform the analysis, we need to
know the joint probability density function of these variables. Very
often the experimental information on the probability densities is
lacking, and the researchers are making unsubstantiated and unverified
assumptions on the density and distribution functions.
The probabilistic analyst in actuality claims: ``Give me the joint
probability densities of random variables involved, and I will calculate
the probability of the mechanical system performing its designated
mission with desired accuracy!" This reminds us of the well-known
statement by Archimedes: ``Give me a firm spot on which to stand, and I
will move the Earth". It should be borne in mind that, as Wentzel
\cite{Wentzel 1988} notes, probabilistic models are ``...viewed to be a
sort of magic wand yielding information from nothing, i.e., from total
ignorance. Those who think so are under a misapprehension since
probability theory is used but to transform data on observed phenomena
to infer the behavior of those which cannot be observed."

\section{Convex Modeling}
Very often, the information on uncertain variables is fragmentary.
Indeterminacy about the uncertain variables involved could be formulated
in terms of these variables belonging to certain sets, such as:
\begin{itemize}
\item[a)]The uncertain parameter $x_j$ is bounded: $$\underline{a}_j\le
x_j\le\overline{a}_j,$$ where $\underline{a}_j$ and $\overline{a}_j$ are
lower and upper values, respectively, that the variable $x_j$ can take.
\item[b)]The uncertain function has envelope bounds:
$$\underline{x}(t)\le x(t)\le\overline{x}(t),$$ where 
$\underline{x}(t)$ and $\overline{x}(t)$ are deterministic functions
which delimit the range of variation of $x(t)$. 
\item[c)]The uncertain function has an integral square bound:
$$\int x^2(t)\,dt\le a.$$
\item[d)]The Fourier coefficients $A$ of the uncertain function $x(t)$
belong to an ellipsoidal set $$A^TWA\le a^2,$$ where $W$ is the weighing
matrix.
\end{itemize}
All above models represent convex sets. As is seen, the information
contents of a convex model is usually a bound or constraint which
defines an infinite set of values of functional forms which an uncertain
variable may assume. Let us consider a set-theoretic approach to
modeling the uncertainty of a time dependent macroscopic vector function
$f$. Let $\Gamma$ be a set of vector-values functions. For a positive
integer $n$, consider the set of functions:
$$F_n=\{f\,|\,f(t)={1\over n}\sum_{i=1}^n g_i(t)\ {\rm for}\
g_i\in\Gamma, i=1,2,...,n\}.$$
$F_n$ is the set of $n-$fold averages of vector functions in $\Gamma$.
It is known \cite{Artstein 1974} that as $n\to\infty$, the sequence of
sets $F_n$ converges to the convex hull of $\Gamma.$ Or, in physical
terms, if a macroscopic time-dependent vector $f(t)$ is formed as the
superposition of numerous microscopic time-varying events $g_i(t)$
chosen from a set $\Gamma$, then the set of all such functions $f(t)$
will tend to be convex, regardless of the structure of the set $\Gamma$.

The convex modeling of uncertainty produces:
\begin{itemize}
\item  maximum, or least favorable
response of the structure, and 
\item
minimum, a best favorable response. 
\end{itemize}
It was applied to:
\begin{itemize}
\item vehicles moving on the rough surface with bounded terrain
\cite{Elishakoff Ben-Haim 1990,Elishakoff Cai Starnes 1994};
\item stress concentration in elastic sheets with bounded irregularities
\cite{Givoli Elishakoff 1992};
\item optimization of trusses \cite{Elishakoff Haftka Fang 1994},
\item and several other problems.
\end{itemize}

\section{Convex Modeling Represents a Generalization of Interval
Analysis}
Indeed, type a) of the above four constraints is a representation of an
interval variable. For this type of convex models, one can utilize the
ample knowledge accumulated in interval analysis:
\begin{itemize}
\item 
Elishakoff and Duan  \cite{Elishakoff Duan 1994} have utilized interval
analysis to deal with interval masses and stiffnesses of the system.
\item Analogously, Qiu et al \cite{Qiu et al 1992,Qiu et al 1994} developed
interval results for natural frequencies, whereas 
\item K\"oyl\"uoglu et al \cite{K et al 1994} developed interval
finite element analysis.
\end{itemize}

The lecture presents some recent developments of convex (and, in
particular, of interval) analysis of structures to analyze uncertainty,
and future research needs. 

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\end{thebibliography}

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