MinimumBias_journal-published

MinimumBias_journal-published - Research Paper Struct...

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Research Paper DOI 10.1007/s00158-004-0433-9 Struct Multidisc Optim 28, 231–242 (2004) New formulation of minimum-bias central composite experimental design and Gauss quadrature ? X. Qu, G. Venter and R.T. Haftka Abstract Response surface methods provide a powerful toolforconstructingapproximationstocomplexresponse functions. Statistical design of experiments is usually used to select optimal points that minimize the error in the resulting response surface approximation. Tradition- ally, data points are selected using minimum-variance designs, for example the D-optimal design, which may result in large bias errors for low-order approximation. Minimum-bias criteria have been developed for select- ing data points to minimize the bias error of a response surface approximation. The present work developed a minimum-bias counterpart to the popular minimum- variance central composite designs. In addition, a new formulation of the minimum-bias design that assigns unequal weights to the design points, based on Gauss quadrature,is explored.Example problems areevaluated and the results obtained from D-optimal, the traditional minimum-bias, and the new Gauss-quadrature-based minimum-bias designs arecompared.It is shown that the Gauss-quadrature-based minimum-bias design criterion resultsin the mostaccurateapproximationsandprovides analytical solutions to a wider range of approximation domains than the traditional minimum-bias design. Re- sponse surface approximations based on minimum-bias central composite designs are more accurate than those constructed from traditional central composite design. Moreover, it is shown that using weights in regression has little influence on the accuracy of the response sur- Received: 28 July 2003 Revised manuscript received: 16 April 2004 Published online: 27 July 2004 Springer-Verlag 2004 X. Qu 1 , ) ,G .Venter 2 and R.T. Haftka 1 1 Dept. of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL32611-6250, USA e-mail: xueyong@ufl.edu, haftka@ufl.edu 2 Vanderplaats Research and Development, Inc, 1767 S 8th Street,Suite200, Colorado Springs,CO 80906, USA e-mail: gventer@vrand.com ? Preliminary version of the paper presented at the Fifth World Congress on Structural and Multidisciplinary Opti- mization, Venice, Italy, May 19–23, 2003 face approximation in Gauss-quadrature minimum-bias designs. Keywords centralcompositedesign,Gaussquadrature, minimum-bias designof experiment, response surfaceap- proximation 1 Introduction Response surface methods are used to construct sim- ple approximations to the response of complex systems. Theseresponsesurfaceapproximationsareusedinalarge numberofproblemareas,includingengineeringoptimiza- tion and reliability evaluations (Kaufman etal. 1996; Venter 1997; Qu 2003). Response surface ap- proximations are generally smooth, low-order polynomi- alsand havethe desirablepropertyof eliminating numer- icalnoise,whichisinherentinmostengineeringcomputer simulations (e.g. Giunta 1994; Venter 1997).
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MinimumBias_journal-published - Research Paper Struct...

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