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diff --git a/macros/latex/contrib/springer/svjour/stropt/example/example.tex b/macros/latex/contrib/springer/svjour/stropt/example/example.tex new file mode 100644 index 0000000000..e5e2cb300e --- /dev/null +++ b/macros/latex/contrib/springer/svjour/stropt/example/example.tex @@ -0,0 +1,1210 @@ +\documentclass[stropt]{svjour} +%\usepackage{times} +\usepackage{graphicx} +% +%Definitions +% +\def\be{\begin{equation}}% +\def\ee{\end{equation}}% +\def\m{\mathbf{M}}% +\def\b{\mathbf{B}}% +\def\y{\mathbf{Y}}% +\def\x{\mathbf{X}}% +\def\p{\mathbf{P}}% +\def\xx{\mathbf{x}}% +\def\zz{\mathbf{z}}% +\def\bb{\mathbf{\beta}}% +\def\kb{\mathbf{b}}% +% +\begin{document} +% +\title{Using response surface approximations\\ in fuzzy set based +design optimization\thanks{Presented as paper 98--1776 at the +39th AIAA\-/ASME/\-ASCE/\-AHS/\-ASC Structures, Structural Dynamics, and Materials +Conference, Long Beach, California, April 20-23, 1998}} +\author{G. Venter and R.T. Haftka} +% +\institute{Department of Aerospace Engineering, Mechanics and Engineering +Science, University of Florida, Gainesville, Florida 32611--6250, USA\\ +\email{gventer@ufl.edu} and \email{haftka@ufl.edu}} +% +\date{Received: August 24, 1998} +% The correct dates will be entered by Springer +% +\maketitle +% +\begin{abstract} +The paper focuses on modelling uncertainty typical of the aircraft industry. +The design problem involves maximizing a safety measure of an isotropic +plate for a given weight. +Additionally, the dependence of the weight on the level of uncertainty, +for a specified allowable possibility of failure, is also studied. +It is assumed that the plate will be built from future materials, with +little information available on the uncertainty. +Fuzzy set theory is used to model the uncertainty. +Response surface approximations that are accurate over the entire design +space are used throughout the design process, mainly to reduce the +computational cost associated with designing for uncertainty. +All of the problem parameters are assumed to be uncertain, and both a +yield stress and a buckling load constraint are considered. +The fuzzy set based design is compared to a traditional deterministic +design that uses a factor of safety to account for the uncertainty. +It is shown that, for the example problem considered, the fuzzy +set based design is superior. +Additionally, the use of response surface approximations results in +substantial reductions in computational cost, allowing the final results to +be presented in the form of design charts. +\end{abstract} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{Introduction} +\label{sec01} + +In the aircraft industry, structures are often designed that will be built +well into the future from materials available then, leading to uncertainty +in material properties. +Apart from the uncertain material properties, the manufacturing cost is also +uncertain. +However, unlike the uncertainty in the material properties, the designer has +some control over the manufacturing cost, which is closely linked to the +required tolerances in geometry. +For such design problems, little information regarding the uncertainty is +known, and the uncertainty is typically modeled based on expert opinion and +assumptions made by the designer. +Fuzzy set theory can use limited available data and caters for worst case +scenarios. +Fuzzy set theory is thus capable of by compensating for the fact that the +uncertainty is modeled based on subjective opinions and assumptions +\citep{Maglaras97}. +In contrast, probabilistic methods require large amount of data and the +results obtained are, in some cases, very sensitive to both the accuracy of +this data as well as to the assumptions made during the modeling process +\citep[e.g.][pp.~11--32]{BenHaim90}. + +Fuzzy set theory was introduced by \citet{Zadeh65} +as a mathematical +tool for the quantitative modeling of uncertainty, and makes use of fuzzy +numbers to represent uncertain problem parameters. +The designer only needs to specify the range of uncertainty and a membership +function that denotes the possibility of occurrence of an element in the +specified range to represent an uncertain parameter as a fuzzy number. +Membership functions are generally constructed subjectively, based on +expert opinion. +In recent years, fuzzy set theory has been applied to a wide range of +structural optimization problems. +For example, \cite{Liu92} +performed a fatigue reliability +analysis of a portal frame, \cite{Jung96} +considered the +optimal plastic design of a fixed-fixed beam and a portal frame, and +\cite{Jensen97} +minimized the weight of a 25-bar transmission tower. +Fuzzy set theory has also been used in multidisciplinary optimization by +\cite{Rao93} +to design the main rotor of a helicopter as well as by +\cite{Wu96} +to optimize the machine room layout of a ship. +Additionally, \cite{Shih95} +applied multicriteria +optimization to various truss examples, considering both weight and +displacement as objectives. + +Unfortunately, designing for uncertainty is computationally intensive and +typically requires at least an order of magnitude more computational cost as +compared to a corresponding deterministic design. +In the present paper, response surface approximations are used to reduce the +high computational cost associated with designing for uncertainty by using +approximations that are accurate over the entire design space to replace +costly finite element analyses. +Response surface approximations have attracted a lot of interest from the +structural optimization community in recent years, since they filter out +numerical noise inherent to most numerical analysis procedures +\citep[e.g.][]{Giunta94}, +they provide the designer with a global +perspective of the response over the entire design space +\citep[e.g.][]{Mistree94}, +and they enable easy integration of +various software codes \citep[e.g.][]{Kaufman96}. + +An isotropic plate with a change in thickness across its width is considered +as a design problem. +All of the problem parameters are considered to be uncertain and the +objective is to maximize a safety measure of the plate for a given weight. +Both deterministic and fuzzy set based designs are considered and the results +are compared. +The safety measure is maximized by maximizing the factor of safety in the +deterministic design and by minimizing the possibility of failure in the +fuzzy set based design. +Finally, the dependence of the weight on the level of uncertainty +associated with the key geometric parameters is presented in the form of a +design chart, based on results obtained from a number of optimizations. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\section{Fuzzy set theory} +\label{sec02} + +Fuzzy set theory presents a methodology for the mathematical modeling of +uncertainty. +In contrast to classical set theory where a sharp transition exists between +membership and non-membership, fuzzy set theory makes use of membership +functions to denote the degree to which an element belongs to a fuzzy set. +A membership function assigns a grade of membership, ranging between 0 and 1, +to each element of the universal set as follows +\begin{equation} +\m (x):~X\rightarrow [0,1]\, . \label{eq01} +\end{equation} +In (\ref{eq01}) $\m$ denotes a membership function that maps the elements of +the universal set $X$ to the real interval $[0,1]$. +The same symbol, a bold face capital letter, is used to denote both the fuzzy +set and its membership function. +Since each fuzzy set is completely and uniquely defined by only one particular +membership function, no ambiguity results from the double use of the symbol. + +Fuzzy sets are represented numerically by making use of $\alpha$ level cuts. +An $\alpha$ level cut is defined as the real interval where the membership +function is larger than a given value, $\alpha$ \citep[p.~19]{Klir95} +and may be written mathematically for a generic +fuzzy set $\b$ as follows: +\begin{equation} +^{\alpha}B=\left\{ x | \b (x)\geq\alpha\right\}\, . \label{eq02} +\end{equation} +Figure~\ref{fig:1} provides a graphical representation of (\ref{eq02}), where +it is assumed that $\b$ has a triangular and symmetric membership function, +and shows the end points $^{\alpha}b_1$ and $^{\alpha}b_2$ of the $\alpha$ +level cut. + +\begin{figure} +\vspace{5cm} +\caption{An $\alpha$ level cut of a triangular and symmetric membership +function, having support in $(x_L, x_R)$} +\label{fig:1} +\end{figure} + +A fuzzy number is defined as a fuzzy set that is both normal and convex +\citep[pp.~97]{Klir95}. +A normal fuzzy set has a maximum membership function equal to 1, while all +possible $\alpha$ level cuts are convex for a convex fuzzy set. +The fuzzy set $\b$ +shown in Fig.~\ref{fig:1} is thus a fuzzy number. +In fact the triangular and symmetric membership function is most often used to +represent fuzzy numbers, mainly due to its simplicity, and was used throughout +the present paper to represent all of the uncertain problem parameters. + +A fuzzy function $\y$ is a function of fuzzy variables $\x_i$ and may be +written as +\begin{equation} +\y=\y (\x_1,\x_2,\ldots,\x_n) \label{eq03} +\end{equation} +for the case where $n$ fuzzy variables are considered. +%Klir and Yuan 1995, +%(pp.~105--109) +\citet[pp.~105--109]{Klir95} summarized and proved the +following properties of a fuzzy function. + +\begin{enumerate} +\item When all of the fuzzy variables of a fuzzy function are continuous +fuzzy numbers, the fuzzy function itself is also a continuous fuzzy number. +\item When all of the fuzzy variables of a fuzzy function are fuzzy numbers, +the $\alpha$ level cut of a fuzzy function $^{\alpha}Y$ may be written in +terms of the $\alpha$ level cuts of its fuzzy variables $^{\alpha}X_i$ as +follows: + +\begin{gather} +^{\alpha}Y=^{\alpha}Y(^{\alpha}X_1,%^{\alpha}X_2, +\ldots,^{\alpha}X_n)=\label{eq04}\\ +\left[ +\min_{^{\alpha}R} +\left[ +Y(^{\alpha}X_1,%^{\alpha}X_2, +\ldots,^{\alpha}X_n) +\right], +\max_{^{\alpha}R} +\left[ +Y(^{\alpha}X_1,%^{\alpha}X_2, +\ldots,^{\alpha}X_n) +\right] +\right]\, ,\notag +\end{gather} +where $^{\alpha}R$ denotes the $n$-dimensional box, formed by the $\alpha$ +level cuts of the $n$ fuzzy numbers. +\end{enumerate} +Based on these properties of a fuzzy function, +\cite{Dong87} +introduced the vertex method for evaluating the upper and lower bounds of +$^{\alpha}Y$ when all of the fuzzy variables of $\y$ are fuzzy numbers. +This method requires the evaluation of the fuzzy function at the $2^n$ +vertices of the $n$-dimensional rectangle, formed by the $\alpha$ level cuts +of the $n$ fuzzy variables. +In addition, interior global extreme points need to be checked. +This method requires a large number of function evaluations and is +computationally intensive. + +For calculating the possibility of failure it is required to compare a crisp +number with a fuzzy number. +Note that a fuzzy number may also be considered as the trace of a possibility +measure $\Pi$ on the singletons (single elements) $x$ of the universal set $X$ +%(Dubois and Prade 1988, +%pp.~13--17). +\citep[p.~13--17]{Dubois88}. +When a possibility measure defined on the unit interval is considered, its +possibility distribution $\pi$ is then interpreted as the membership function +of a fuzzy number $\b$ describing the event that $\Pi$ focuses on, as follows: +\begin{equation} +\Pi\left( \{ x\}\right)=\pi(x)=\b (x)\, ,\quad \forall x\in X\, . \label{eq05} +\end{equation} +The possibility measure of a crisp number being smaller or equal to a fuzzy +number $\b$ is then defined \cite[pp.~99--101]{Dubois88} as follows: +% +\begin{equation} +\Pi_{\b}\left( [x,+\infty) \right)=\sup_{y\geq x}\b (y)\, ,\quad \forall +x\, . \label{eq06} +\end{equation} +The possibility distribution function $\pi_{\b}$ corresponding to +the possibility measure of (\ref{eq06}) is shown graphically in +Fig.~\ref{fig:2} for the general case where $\b$ has a nonlinear membership +function. + +\begin{figure} +\vspace{5cm} +\caption{Possibility distribution of $\b \geq x$ for nonlinear $\b(x)$, +having support in $(x_L, x_R)$} +\label{fig:2} +\end{figure} + +Based on (\ref{eq05}) and (\ref{eq06}), the possibility distribution of +failure $\pi_{(\p-\p_f)}$ is obtained from the fuzzy function +$(\p-\p_f)$ that contains the fuzzy numbers $\p$ (the applied load) and $\p_f$ +(the failure load) as variables. +The possibility of failure $(\p-\p_f\geq 0)$ is then defined as +\begin{equation} +\Pi_{(\p-\p_f)}\left( [0,+\infty)\right)=\sup_{y\geq 0}(\p-\p_f)(y)\, . +\label{eq07} +\end{equation} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\section{Overview of response surface approximations} +\label{sec03} + +A response surface approximation is an approximate relationship between a +dependent variable $\eta$ (the response) and a vector $\xx$ of $k$ +independent variables (the predictor variables). +The response is generally obtained from experiments (which may be numerical +in nature), where $\eta$ denotes the mean or expected response value. +It is assumed that the true model of the response may be written as a linear +combination of given functions $\tilde{\zz}$ with some unknown coefficients +$\tilde{\bb}$. +The experimentally obtained response $y$ differs from the expected value +$\eta$ due to random experimental error $\delta$ as follows: +\begin{equation} +y(\xx)=\eta(\xx)+\delta=\tilde{\zz}(\xx)^T \tilde{\bb}+\delta\, . +\label{eq08} +\end{equation} + +Since the exact dependence of $\eta$ is generally unknown, a response surface +approximation is used to approximate $\eta(\xx)$ as follows: +\begin{equation} +y(\xx)=\zz(\xx)^T\bb+\varepsilon\, , +\label{eq09} +\end{equation} +where $\zz(\xx)$ contains the assumed functions in the response surface +approximation and $\bb$ the associated coefficients. +Furthermore, $\varepsilon$ denotes the total error, which is the +difference between the predicted and measured response values and includes +both random (variance) and modeling (bias) error. +Typically low order polynomials are used as a response surface approximation, +in which case $\zz(\xx)$ consists of monomials. + +The coefficients $\bb$ of the response surface approximation are estimated +from the experimentally obtained response values to minimize the sum of the +squares of the error terms, a process known as regression. +The estimated values of $\bb$ is denoted by $\kb$, resulting in the +following response surface approximation: +\begin{equation} +\hat{y}(\xx)=\zz(\xx)^T\kb\, , +\label{eq10} +\end{equation} +where the caret symbol implies predicted values. + +The assumed form of the response surface approximation, (\ref{eq10}), +usually includes redundant parameters and parameters that are poorly +characterized by the experiments. +These parameters may increase the prediction error of the approximation and +thus decrease its predictive capabilities. +In the present paper, redundant parameters are eliminated by using mixed, +backwards, stepwise regression (e.g. +\citealt{Ott93}, pp.~648--659; \citealt{Myers95}, pp.~642--655). +Mallow's $Cp$ statistic is used to identify the best reduced response +surface approximation from the subset of reduced response surface +approximations provided by the stepwise regression procedure and is +defined as +\begin{equation} +Cp=\frac{SSE_p}{s_{\varepsilon}^2}-(n-2p)\, , +\label{eq11} +\end{equation} +where $SSE_p$ is the sum of the squares of the $n$ error terms (one for each +data point used to estimate $\kb$) for an approximation with $p$ parameters +and $s_{\varepsilon}^2$ is the mean sum of squares of the error terms obtained +from the response surface approximation with all of the parameters included. + +Optimization has the general tendency of exploiting weaknesses in the +formulation of the response function, and highly accurate response surface +approximations are thus a requirement in structural optimization applications. +To ensure highly accurate approximations, it is important to evaluate the +predictive capabilities of the approximations. +In the present paper, the coefficient of determination $(R^2)$ statistic, +the adjusted $R^2$ (Adj-$R^2$) statistic, the percent root mean square +error (\%RMSE) as well as the percent root mean square error based on the +predicted sum of squares (PRESS) statistic (\%RMSE$_{{\rm PRESS}}$) are +calculated \citep[pp.~28--47]{Myers95}. + +The $R^2$ statistic denotes the proportion of the variability in the response +that is accounted for by the response surface approximation and has a value +between 0 and 1. +The Adj-$R^2$ statistic is an alternative measure of the explained variability +that, unlike $R^2$, has the desirable property that its value does not +necessarily increase when adding (possibly redundant) parameters to a +response surface approximation. +The \%RMSE is an estimate of the root mean square error of the approximation +that is obtained from the data points used to construct the approximation, +using the following unbiased estimator: +% +\[ +{\rm \%RMSE}=\frac{100}{\overline{y}}=\sqrt{\frac{1}{(n-p)} +\sum\limits_{i=1}^{n}(y_i-\hat{y}_i)^2}\, , +\] +where +% +\begin{equation} +\overline{y}=\frac1n\sum\limits_{i=1}^{n}| y_i |\, . +\label{eq12} +\end{equation} + +The \%RMSE$_{{\rm PRESS}}$ is an additional measure of the error, based on +the PRESS statistic. +The PRESS statistic is calculated by selecting a data point, say data point +$i$. +The response surface approximation obtained from the remaining $(n - 1)$ data +points is used to predict the response at the withheld data point, denoted by +$\hat{y}_{(i)}$. +The prediction error at the withheld data point $e_{(i)}$ is then defined as +\begin{equation} +e_{(i)}=y_i-\hat{y}_{(i)}\, , \label{eq13} +\end{equation} +and is referred to as the $i$-th PRESS residual. +This procedure is repeated for all of the data points and the resulting PRESS +residuals are summed to form the PRESS statistic as follows: +\begin{equation} +{\rm PRESS}=\sum\limits_{i=1}^{n}e^2_{(i)} +=\sum\limits_{i=1}^{n}\left\lbrack y_i-\hat{y}_{(i)}\right\rbrack ^2\, . +\label{eq14} +\end{equation} +The %RMSE$_{{\rm PRESS}}$ is then defined as: +\begin{equation} +{\rm \%RMSE}_{{\rm PRESS}}=\frac{100}{\overline{y}} +\sqrt{\frac1n{\rm PRESS}}\, . +\label{eq15} +\end{equation} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\section{Plate example} +\label{sec04} + +An isotropic plate with a change in thickness in the form of a linear ramp +(see Fig.~\ref{fig:3}) is the design problem considered in the present paper. + +\begin{figure} +\vspace{5cm} +\caption{Three-dimensional view of the plate with a thickness change.} +\label{fig:3} +\end{figure} + +Three nondimensional parameters, $\lambda$, $\beta$ and $\gamma$, are used to +specify the geometry and location of the change in thickness +(see Fig.~\ref{fig:4}). +The plate is simply supported on two edges, free on the other two edges, and +subjected to an uniformly distributed load applied on the two simply supported +edges. + +\begin{figure} +\vspace{5cm} +\caption{Cross-section of plate with response variables shown.} +\label{fig:4} +\end{figure} + +Both a yield stress failure (according to the Von Mises criterion) and a +buckling load constraint are considered in the design, and the failure load +$P_f$ of the plate is calculated from +\begin{equation} +P_f=\min\left\{ +\begin{array}{l} +\frac{\strut\displaystyle\sigma_Y\lambda b t_0}{\strut\displaystyle\tilde{\sigma}_x} \\ +\frac{\strut\displaystyle\tilde{N}_{{\rm crit}} \pi^2 Eb(\lambda t_0)^3}{\strut\displaystyle12(1-\nu^2)a^2} +\end{array} +\right.\, , \label{eq16} +\end{equation} +and failure is defined to occur when: +\begin{equation} +P-P_f\geq 0\, . \label{eq17} +\end{equation} +In (\ref{eq16}) and (\ref{eq17}), $\sigma_y$ denotes the yield stress, $E$ the +Young's modulus and $\nu$ the Poisson's ratio of the material considered, +while $\lambda$, $a$, $b$, $t_0$ and $r$ describe the geometry of the plate as +shown in Figs.~\ref{fig:3} and \ref{fig:4} and $P$ denotes the applied load. +Additionally, $\tilde{\sigma}_x$ denotes the nondimensional, $x$-directional +stress component on the top surface of the thin section of the plate, +calculated a distance $r$ from the re-entrant corner, and is defined as +\begin{equation} +\tilde{\sigma}_x=\frac{\lambda b t_0\sigma_x}{P}\, , +\label{eq18} +\end{equation} +while $\tilde{N}_{{\rm crit}}$ denotes the nondimensional buckling load of +the plate, defined as +\begin{equation} +\tilde{N}_{{\rm crit}}=\frac{12(1-\nu^2)a^2 N_{{\rm crit}}}{\pi^2 E b +(\lambda t_0)^3}\, . +\label{eq19} +\end{equation} + +\leavevmode\citet{Venter97} +used a large number of numerical experiments to +study this problem in detail, and showed that the maximum von Mises stress +always occurs on the top surface of the thin section of the plate, in which +case $\sigma_x$ is the only nonzero stress component. +According to the von Mises criterion, failure then occurs when +\begin{equation} +\tilde{\sigma}_x\geq \tilde{\sigma}_Y=\frac{\lambda b t_0\sigma_Y}{P}\, . +\label{eq20} +\end{equation} +\cite{Venter97} +also determined that the problem has both a local +and a global buckling mode, and defined a simple geometric criterion to +distinguish between the two buckling modes as follows: +\begin{equation} +\mbox{buckling mode}=\left\{ +\begin{array}{ll} +\mbox{local if} & \frac{(0.5-\beta-\gamma)}{\lambda}\geq 0.6 \\[4pt] +\mbox{global if} & \frac{(0.5-\beta-\gamma)}{\lambda}\leq 0.6 +\end{array}\, . +\right. \label{eq21} +\end{equation} + +\leavevmode\cite{Venter97} +constructed highly accurate response surface +approximations for both the $x$-directional stress distribution on the top +surface of the thin section of the plate and for the buckling load of the +plate, using a total of 752 finite element analyses. +Numerical experiments in the form of finite element analyses were conducted +using MSC$\backslash$NASTRAN Version 68. +A cross-section of the plate was used to model the stress distribution near +the re-entrant corner, using four-node, isoparametric, plane strain elements. +All of these models had a uniform mesh, with roughly 1,800 elements in the +$x$-direction and 9 elements in the $z$-direction, but the number of elements +varied slightly from model to model. +A schematic representation of the finite element model used is shown in +Fig.~\ref{fig:5}. + +\begin{figure} +\vspace{5cm} +\caption{Finite element model used for stress distribution about the +re-entrant corner} +\label{fig:5} +\end{figure} + +For the buckling load response surface approximations, four-node, +isoparametric, plate bending elements were used to construct a two-dimensional +finite element model similar to a plan view of Fig.~\ref{fig:3}. +Twenty elements were used in each of the $x$- and $y$-directions respectively. +The eccentricity of the mid-plane was found to have an insignificant impact on +the buckling load value (note that the sides of the plate are free) and was +ignored in the analysis. + +The stress distribution response surface approximation +\citep[see][]{Venter97} +%(see +%\cite{Venter97}) +may be written in functional form as +\begin{equation} +\tilde{\sigma}_x=\tilde{\sigma}_x\left(\lambda,\beta,\gamma, +\tilde{r}^{\zeta-1}\right)\, , +\label{eq22} +\end{equation} +where $\zeta$ is a constant that describes the radial stress distribution near +the re-entrant corner and depends on $\lambda$, $\gamma$ and $a/t_0$ through +the angle $\Theta$. +Additionally, $\tilde{r}$ is the nondimensional distance measured from the +re-entrant corner, defined as: +\begin{equation} +\tilde{r}=r/t_0\, . +\label{eq23} +\end{equation} +Additionally, two response surface approximations, corresponding to the local +and global buckling modes were constructed, which may be written in functional +form as: +\begin{equation} +\tilde{N}_{{\rm loc}}=\tilde{N}_{{\rm loc}}(\lambda,\beta,\gamma)\, , +\quad +\tilde{N}_{{\rm glob}}=\tilde{N}_{{\rm glob}}(\lambda,\beta,\gamma)\, . +\label{eq24} +\end{equation} + +The design space used for constructing the stress distribution and buckling +load response surface approximations is summarized in Table~\ref{tab:1}. +The upper limit on $\tilde{r}$ limits the radius of the yield zone about the +re-entrant corner to be no greater than 80\% of the thickness of the thin +section of the plate, while the upper bound on $\gamma$ is dictated by the +geometry of the transition region. + +\begin{table}[htbp] +\caption{Design space for constructing the response surface approximation +approximations\hsize=164pt} +\label{tab:1} +\tabcolsep=10pt +\begin{tabular}{@{}ll@{}} +\hline\noalign{\smallskip} +Response variable & Range \\ +\noalign{\smallskip}\hline\hline\noalign{\smallskip} +$\lambda$ & $0.2\leq \alpha\leq 1.0$ \\ +$\beta$ & $-0.475\leq \beta\leq 0.475$ \\ +$\gamma$ & $0\leq \gamma\leq 0.475-\beta$ \\ +$\tilde{r}^{\zeta-1}$ & $0\leq \tilde{r}\leq 0.8\alpha$ \\ +\noalign{\smallskip}\hline +\end{tabular} +\end{table} + +The stress distribution response surface approximation was constructed from +288 plate configurations (corresponding to 288 finite element analyses). +Each plate configuration included a number of data points with different +$\tilde{r}^{\zeta-1}$ values (corresponding to different finite elements), +yielding a total of 2,124 data points. +The buckling load response surface approximations were constructed from an +additional 288 finite element analyses. +Using the geometric criterion of (\ref{eq21}), these 288 finite element +analyses were divided into two groups corresponding to the two buckling modes. +This process identified 126 data points for constructing the local buckling +load approximation and 162 data points for constructing the global buckling +load approximation. +A quartic polynomial was used as initial response surface approximation for +both the stress distribution and the global buckling load response surface +approximations, while a cubic polynomial was used for the local buckling load +response surface approximation. +These initial response surface approximations were reduced, using the mixed +stepwise regression procedure and the $Cp$ statistic. +The process of constructing the response surface approximations is discussed +in more detail by \cite{Venter97}. +The predictive capabilities of the reduced response surface approximations are +summarized in Table~\ref{tab:2}. + +\begin{table}[htbp] +\caption{Predictive capabilities of stress distribution and buckling load +response surface approximations\hsize=178pt} +\label{tab:2} +\tabcolsep5pt +\begin{tabular}{@{}lcccc@{}} +\hline\noalign{\smallskip} +{Model} & ${\mathbf{R^2}}$ & {Adj-}$\mathbf{R^2}$ &{RMSE}&{PRESS}\\ +&&&{[\%]}&{[\%]} \\ +\noalign{\smallskip}\hline\noalign{\smallskip} +{Stress} & \multicolumn{4}{c}{{4-th order model (2,124 data + points)}} \\ +\noalign{\smallskip}\hline\noalign{\smallskip} +Reduced\\ +43 terms& 0.9983 & 0.9982 & 3.2964 & 3.3886 \\ +{Local}\\ +{buckling}&\multicolumn{4}{c}{{3-rd order model (126 data points)}} \\ +\noalign{\smallskip}\hline\noalign{\smallskip} +Reduced\\ +19 terms& 0.9999 & 0.9998 & 0.5550 & 0.6920 \\ +{Global}\\ + {buckling}& \multicolumn{4}{c}{{4-th order model (162 data points)}} \\ +\noalign{\smallskip}\hline\noalign{\smallskip} +Reduced\\ + 25 terms& 0.9910 & 0.9895 & 2.4888 + & 3.0202 \\ +\noalign{\smallskip}\hline +\end{tabular} +\end{table} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\section{Design problem formulation} +\label{sec05} + +The design problem has two objectives. +The first objective is to maximize a safety measure of the plate for a given +weight. +The results obtained from a traditional deterministic approach, using a factor +of safety to account for the uncertainty, were compared to those obtained from +a fuzzy set based approach. +The safety measure of the plate was maximized, by maximizing the factor of +safety for the deterministic approach and by minimizing the possibility of +failure for the fuzzy set based approach. +Note that there exist fundamental differences between the deterministic and +fuzzy set based approaches for maximizing the safety measure of the plate for +a given weight. +The deterministic approach tends to equalize the failure load of each failure +criterion, while the fuzzy set based design tends to equalize the possibility +of failure of each failure criterion. + +The second objective is to study the dependence of the weight of the final +design on the level of uncertainty associated with the design variables +$\lambda$, $\beta$ and $\gamma$. +In this case, the weight was minimized for a specified allowable possibility +of failure and different levels of uncertainty associated with the design +variables. +The results are presented in the form of a design chart. +Different levels of uncertainty for the design variables were considered, +since these geometric variables have the largest influence on the +manufacturing cost of the plate. +If the tolerances of these variables can be relaxed without a large penalty +in terms of weight, substantial cost savings can be achieved in manufacturing +the plate. +The problem parameters and associated levels of uncertainty used are +summarized in Table~\ref{tab:3}. +Although Table~\ref{tab:3} has a total of 11 uncertain problem parameters, +only 8 uncertain parameters are associated with each of the two failure +criteria [see (\ref{eq16}), (\ref{eq22}) and (\ref{eq24})]. + +\begin{table}[htbp] +{\hsize=156pt +\caption{Problem parameters and associated uncertainty}} +\label{tab:3} +\tabcolsep5pt +\begin{tabular}{@{}lcc@{}} +\hline\noalign{\smallskip} +Variable & \begin{tabular}{c}Nominal \\ values \end{tabular} & + \begin{tabular}{c}Level of \\ uncertainty, $\mathbf{u}$ \end{tabular} \\ +\noalign{\smallskip}\hline\hline\noalign{\smallskip} +$\lambda^{\dagger}$ & [0.2 -- 1.0] & [$\pm$ 2 -- $\pm$ 20]\% \\ +$\beta^{\dagger}$ & [-0.4 -- 0.4] & [$\pm$ 2 -- $\pm$ 20]\% \\ +$\gamma^{\dagger}$ & [0 -- 0.8] & [$\pm$ 2 -- $\pm$ 20]\% \\ +$a$ & 228.6~cm & $\pm$5\% \\ +$b$ & 127.0~cm & $\pm$5\% \\ +$t_0$ & 7.620~cm & $\pm$5\% \\ +$E$ & 206.84~GPa & $\pm$5\% \\ +$\nu$ & 0.29 & $\pm$5\% \\ +$\sigma_y$ & 197.26~MPa & $\pm$10\% \\ +$r$ & $5\alpha t_0$ & $\pm$10\% \\ +$P$ & 3,224.96~kN & $\pm$10\% \\ +\noalign{\smallskip}\hline +\end{tabular} + +\noindent +{\footnotesize $^{\dagger}$ Design variables} +\end{table} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\subsection{Deterministic design} +\label{sec05.01} + +The objective of the deterministic design is to maximize the factor of safety +for a given weight. +However, since it is difficult to specify a meaningful weight, it was decided +to minimize the weight for a given factor of safety. +The resulting minimum weight was then used as the given weight for the fuzzy +set based design. +A factor of safety of 1.5 was assumed and the level of uncertainty associated +with the design variables was considered to be constant, equal to $\pm$5\%. +The nondimensional cross-sectional area of the plate $\tilde{A}$ was used as +a representative value of the weight and the resulting optimization problem +may be written as\\[6pt] +minimize: +\[ +\tilde{A}=\frac{A}{\lambda t_0}=\frac12 (1+2\beta+\gamma)+\frac{\lambda}{2} +(1-2\beta-\gamma)\, , +\] +subject to +\begin{gather} +\frac{\beta}{0.4}+1\geq 0\, ,\quad +1-\frac{\beta}{0.4}\geq 0\, ,\quad +\gamma\geq 0\, ,\notag\\ +1-\frac{\gamma+\beta}{0.4}\geq 0\, ,\quad +\frac{P_f}{P}-1.5\geq0\, . +\label{eq25} +\end{gather} +The constraints involving $\beta$ and $\gamma$ are geometric constraints and +$P_f$ is calculated from (\ref{eq16}), using the nominal values of the design +variables. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\subsection{Fuzzy set based design} +\label{sec05.02} + +The fuzzy set based design problem minimizes the possibility of failure, using +the optimum nondimensional cross-sectional area obtained from (\ref{eq25}) as +an upper limit of the weight. +The resulting optimization problem may be written as\\[6pt] +Minimize: +\[ +\Pi_{(\p-\p_f)}=\Pi_{(\p-\p_f)}({\vec{\lambda}}, {\vec{\beta}}, {\vec{\gamma}} )\, , +\] +subject to +\begin{gather} +\frac{\beta}{0.4}+1\geq 0\, ,\quad +1-\frac{\beta}{0.4}\geq 0\, ,\quad +\gamma\geq 0\, ,\notag\\ +1-\frac{\gamma+\beta}{0.4}\geq 0\, ,\quad +\frac{\tilde{A}(\lambda,\beta,\gamma)}{\tilde{A}^*}-1=0\, . +\label{eq26} +\end{gather} +where bold face Greek symbols denote fuzzy numbers while regular font symbols +denote nominal values. +Additionally,\linebreak[4]$\Pi_{(\p-\p_f)}$ denotes the possibility of +failure and +$\tilde{A}^*$ denotes the optimum nondimensional cross-sectional area +obtained from the deterministic design of (\ref{eq25}). + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\subsection{Implementation of the fuzzy set based design} +\label{sec05.03} + +In the present work response approximations form an integral part of the fuzzy +set based design and two levels of response surface approximations are +employed during the different stages of the design process. +On the first level, the stress distribution and buckling load response +surface approximations (Section~\ref{sec04}) are used to replace +computationally expensive finite element analysis in evaluating the +possibility of failure. +The possibility of failure is calculated from (\ref{eq16}), using the vertex +method. +When considering all of the problem parameters as uncertain, the evaluation of +the possibility of failure for a single $\alpha$ level cut value requires +$2\times 2^8=512$ (recall that each failure criterion has a total of 8 +uncertain problem parameters) finite element analyses when no response surface +approximations are used. +In terms of a single optimization, an estimate of the required number of +finite element analyses required when not using response surface +approximations, is obtained from the product of four numbers as follows: + +\bigskip\noindent +\begin{tabular}{@{}lc@{}} +\tabcolsep5pt +Average number of design & \\ +optimization iterations: & 5 \\ +\\ +Average number of $\Pi_{(\p-\p_f)}$ & \\ +evaluations per iteration: & 6 \\ +\\ +Average number of $\alpha$ level cut & \\ +evaluations per $\Pi_{(\p-\p_f)}$ evaluation: & 5 \\ +\\ +Number of finite element analyses & \\ +per $\alpha$ level cut evaluation of $\Pi_{(\p-\p_f)}$: & 512 \\ +\cline{2-2} +\\ +Total number of finite element analyses & \\ +required per optimization: & 76,800 +\end{tabular} + +\bigskip +In contrast, the stress distribution and buckling load response surface +approximations were constructed from a total of only 752 finite element +analyses. +Additionally, these response surface approximations can be used in multiple +optimizations without the need of performing additional finite element +analyses. + +On the second level, a response surface approximation of the possibility of +failure as a function of the nominal values of the design variables and the +level of uncertainty associated with these variables was constructed. +This second level approximation was constructed to simplify the integration of +the analysis code with the optimization algorithm as well as to eliminate +noise in the response function, thus allowing the use of a derivative based +optimization algorithm. +In the present paper, the generalized reduced gradient algorithm provided +with Microsoft Excel Version 7.0 was used. + +The $\lambda$, $\beta$ and $\gamma$ design space of Table~\ref{tab:1} was used +to construct the possibility of failure response surface approximation, with +numerical experiments conducted at an evenly spaced grid consisting of 11 data +points in each of the $\lambda$, $\beta$ and $\gamma$ directions. +Additionally, seven levels of uncertainty evenly spaced between $\pm$2\% and +$\pm$20\% were considered, yielding a total of 2,629 data points in the design +space. +At each data point the possibility of failure according to each of the two +failure criteria was evaluated. +Two response surface approximations (one for each failure mode) were +constructed using all of the data points with possibility of failure not +equal to either 0 or 1. +This process resulted in 499 data points for constructing the yield stress +failure criterion response surface approximation and 573 data points for the +buckling load constraint failure criterion response surface approximation. +The resulting predicted possibility of failure is then obtained from +\begin{equation} +\hat{\Pi}_{(\p_-\p_f)}=\min\left(\hat{\Pi}_{{\rm Yield Stress}}, +\hat{\Pi}_{{\rm Buckling}} \right)\, .\label{eq27} +\end{equation} + +It was found that a general fourth-order polynomial (70 parameters) gave +accurate approximations for both failure modes. +These general response surface approximations were reduced using the mixed +stepwise regression procedure and the $Cp$ statistic, with the predictive +capabilities of the response surface approximations summarized in +Table~\ref{tab:4}. + +\begin{table}[htbp] +\caption{Predictive capabilities of the possibility of failure +response surface approximations\hsize=197pt} +\label{tab:4} +\tabcolsep5pt +\begin{tabular}{@{}lcccc@{}} +\hline\noalign{\smallskip} +{Model} & $\mathbf{R^2}$ &{Adj-}$\mathbf{R^2}$ & + \begin{tabular}{c} {RMSE} \\ {[\%]} \end{tabular} & + \begin{tabular}{c} {PRESS} \\ {[\%]} \end{tabular} \\ +\noalign{\smallskip}\hline\noalign{\smallskip} +{Stress} & \multicolumn{4}{c}{{4-th order model (499 data + points)}} \\ +\noalign{\smallskip}\hline\noalign{\smallskip} +Full\\ + 70 terms& 0.9988 & 0.9986 & 2.2525 + & 2.6205 \\ +Reduced\\ + 59 terms& 0.9988 & 0.9986 & 2.2371 + & 2.5205 \\ +{Buckling} & \multicolumn{4}{c}{{4-th order model + (573 data points)}} \\ +\noalign{\smallskip}\hline\noalign{\smallskip} +Full\\ +70 terms& 0.9982 & 0.9980 & 2.7342 + & 3.0991 \\ +Reduced\\ +57 terms& 0.9982 & 0.9980 & 2.7118 + & 3.0211 \\ +\noalign{\smallskip}\hline +\end{tabular} +\end{table} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\section{Dependence of the weight on the level of uncertainty} +\label{sec05.04} + +In order to study the dependence of the weight of the plate on the level of +uncertainty associated with the design variables, different levels of +uncertainty between $\pm$2\% and $\pm$20\% were considered. +For each of these levels, the nondimensional cross-sectional area of the +plate was minimized for an allowable possibility of failure. +The allowable possibility of failure (allowable was assumed to be equal to the +optimum value obtained from the fuzzy set based design problem of +(\ref{eq26}). +The resulting optimization problem may be written as\\[6pt] +minimize: +\[ +\tilde{A}=\frac{A}{\lambda t_0}=\frac12 (1+2\beta+\gamma)+\frac{\lambda}{2} +(1-2\beta-\gamma)\, , +\] +subject to +\begin{gather} +\frac{\beta}{0.4}+1\geq 0\, ,\quad +1-\frac{\beta}{0.4}\geq 0\, ,\quad +\gamma\geq 0\, ,\notag\\ +1-\frac{\gamma+\beta}{0.4}\geq 0\, ,\quad +\frac{\hat{\Pi}_{(\p-\p_f)}(\lambda,\beta,\gamma,u)}{\Pi_{{\rm allowable}}} +-1\geq0\, , \label{eq28} +\end{gather} +where $u$ denotes the level of uncertainty associated with the design +variables and $\hat{\Pi}_{(\p-\p_f)}$ denotes the predicted possibility of +failure, obtained from (\ref{eq27}). + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\section{Results} +\label{sec06} + +In order to obtain an upper limit of the weight for the fuzzy set based +design, the deterministic design was evaluated first. +The nondimensional cross-sectional area of the plate was minimized for a +factor of safety equal to 1.5, making use of the formulation of (\ref{eq25}). +The corresponding optimum design is summarized in Table~\ref{tab:5}, +where the values in parentheses are the possibility of failure values obtained +from the reduced possibility of failure response surface approximations. + +\begin{table}[htbp] +\caption{Deterministic optimum (uncertainty of the design variables equal +to $\pm$5\%)\hsize=109pt} +\tabcolsep5pt +\label{tab:5} +\begin{tabular}{@{}lc@{}} +\hline\noalign{\smallskip} +Variable & Value \\ +\noalign{\smallskip}\hline\hline\noalign{\smallskip} +$\lambda$ & 0.6287 \\ +$\beta$ & -0.4000 \\ +$\gamma$ & 0.0447 \\ +$\tilde{A}^*$ & 0.6741 \\ +Factor of safety & 1.5 \\ +$\Pi_{{\rm Yield stress}}$ & \begin{tabular}{c} 0.0977 \\ (0.1181)\end{tabular}\\ +$\Pi_{{\rm Buckling}}$ & \begin{tabular}{c} 0.3411 \\ (0.3331)\end{tabular}\\ +\noalign{\smallskip}\hline +\end{tabular} +\end{table} + +For the deterministic optimum design, both failure criteria are active. +The optimum design corresponds to a plate with a change in thickness that +starts at the minimum allowable distance from the left endpoint of the plate +(see Fig.~\ref{fig:4}) with a very short transition zone (small $\gamma$ +value). +Even though both failure criteria are active for the optimum design, a large +difference exists between the possibility of failure for the two failure +criteria, with the buckling load constraint being critical. +Both the possibility of failure values obtained from the vertex method and +the values obtained from the reduced possibility of failure response surface +approximation are shown. +The accuracy of the reduced possibility of failure response surface +approximation is demonstrated since the difference between the critical +predicted and calculated possibility of failure values at the optimum design +is only 2.3\%. + +The equivalent fuzzy set based design, using the $\tilde{A}^*$ value +of Table~\ref{tab:5} as an upper limit of the weight are summarized in +Table~\ref{tab:6}. +Again, the values in parentheses are the possibility of failure values +obtained from the reduced possibility of failure response surface +approximations. +The fuzzy set based optimum design corresponds to a plate where the change in +thickness starts at the minimum allowable distance from the left endpoint of +the plate with no transition zone ($\gamma$ value equal to 0). +The fuzzy set based design eliminates the weight of the ramp and uses it to +thicken the thin section of the plate. +The result is an increase in the stress concentration and an improvement in +the buckling load of the plate. +The fuzzy set based design thus attempts to equalize the possibility of +failure of the two failure criteria by making the yield stress failure +criterion more critical and the buckling load constraint less critical. +However, for the present example problem, the design variable limits kept +the possibility of failure values from becoming equal at the optimum design. + +\begin{table}[htbp] +\caption{Fuzzy optimum (uncertainty of the design variables equal +to $\pm$5\%)\hsize=109pt} +\label{tab:6} +\tabcolsep5pt +\begin{tabular}{@{}lc@{}} +\hline\noalign{\smallskip} +Variable & Value \\ +\noalign{\smallskip}\hline\hline\noalign{\smallskip} +$\lambda$ & 0.6379 \\ +$\beta$ & -0.4000 \\ +$\gamma$ & 0.0000 \\ +$\tilde{A}^*$ & 0.6741 \\ +Factor of safety & 1.4898 \\ +$\Pi_{{\rm Yield stress}}$ & \begin{tabular}{c} 0.1319 \\ (0.1262)\end{tabular}\\ +$\Pi_{{\rm Buckling}}$ & \begin{tabular}{c} 0.2788 \\ (0.2721)\end{tabular}\\ +\noalign{\smallskip}\hline +\end{tabular} +\end{table} + +For the fuzzy set based design, the factor of safety is not much different +from that of the deterministic design (only 0.7\% lower), however, there is a +large difference in the possibility of failure between the two designs. +The possibility of failure for the fuzzy set based design is 22.3\% lower than +that of the deterministic design. +As before, both the predicted and calculated possibility of failure values +are shown in Table~\ref{tab:6}, with the difference between the critical +values equal to only 2.4\%. + +The possibility distributions of failure for each failure mode of the optimum +designs obtained from the two methods are shown graphically in +Fig.~\ref{fig:6}. +The possibility distributions of Fig.~\ref{fig:6} clearly illustrate the +differences in the way each method maximizes the safety measure for a given +weight as discussed in Section~\ref{sec05}. + +\begin{figure} +\vspace{5cm} +\caption{Possibility distributions of failure for the deterministic and fuzzy +set based optimum designs} +\label{fig:6} +\end{figure} + +An important tool for determining the tolerances to which a structure will be +manufactured, is to know the dependence of the weight on the uncertainty +associated with the geometry of the structure. +The dependence of the weight of the structure on the level of uncertainty +associated with the design variables was thus also studied. +For this study, the possibility of failure was kept constant at the optimum +value obtained from the fuzzy set based design (i.e., 0.2788 as summarized in +Table~\ref{tab:6}), while the level of uncertainty associated with the design +variables $\lambda$, $\beta$ and $\gamma$ was varied between $\pm$2\% and +$\pm$20\%. +Seven levels of uncertainty, evenly distributed between $\pm$2\% and +$\pm$20\%, were considered. +For each of these levels, the reduced possibility of failure response surface +approximations and the Microsoft Excel solver was used to minimize the +nondimensional cross-sectional area for the specified possibility of failure. +As expected, the nondimensional cross-sectional area of the plate increased +with an increase in the level of uncertainty and the results are shown +graphically in Fig.~\ref{fig:7}. + +\begin{figure} +\vspace{5cm} +\caption{Nondimensional cross-sectional area associated with different level +of uncertainty in the design variables} +\label{fig:7} +\end{figure} + +Figure~\ref{fig:7} indicates that the increase in weight is almost linearly +proportional to the increase in the uncertainty associated with the design +variables. +The nondimensional cross-sectional area increased by 10.7\% with an 18\% +increase in the uncertainty associated with the design variables. +Using Fig.~\ref{fig:7} and the dependence of the manufacturing cost on the +tolerance of $\lambda$, $\beta$ and $\gamma$, the designer may determine what +tolerance to use in manufacturing the plate. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\section{Concluding remarks} +\label{sec07} + +It is shown that response surface approximations provide an effective +approach for reducing the computational cost associated with performing a +fuzzy set based design for uncertainty. +The large number of computationally expensive finite element analyses required +to perform the fuzzy set based design is replaced by response surface +approximations that are inexpensive to evaluate. +By using response surface approximations, the computational burden shifts +from the optimization problem to the problem of constructing the response +surface approximations. +Due to the iterative nature of the design process, the fact that response +surface approximations allow multiple optimizations at minimal cost should be +an attractive feature to any designer. +The present paper also made use of response surface approximations to simplify +the integration of the analysis code and the optimization algorithm. + +It was shown that for the same upper limit of the weight, the fuzzy set based +design resulted in an optimum design with a possibility of failure 22.3\% +lower than the corresponding deterministic design. +Additionally, the factor of safety of the fuzzy set based design is only 0.7\% +smaller than that of the deterministic design and for this example problem the +fuzzy set based design is thus clearly superior. +Finally, the 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