AIAA 2000–4754 Comparison of Approximation Models with Merit Functions for Design Optimization
نویسندگان
چکیده
In this work, the use of both a second-order response surface method (RSM) and the Kriging method as approximation models for design optimization are investigated and compared. After validating the accuracy of each method with simple one-and two-dimensional analytic functions, they are applied to two Supersonic Business Jet (SBJ) drag minimization design cases in order to obtain a clear comparison of the accuracy and efficiency of these modeling techniques. A three-dimensional Euler flow solver and an automatic mesh generation capability are used in the design of a SBJ using a variety of geometric shape design parameters. The comparison results show that there is little difference in modeling accuracy and efficiency between the two methods. In addition, we find that both methods are practically applicable to realistic design optimization problems. Second-order response surface models have a severe limitation in the fact that the model of the function of interest is not a good representation for functions with multiple local minima. Although the Kriging method has the flexibility to capture multiple extrema, it exhibits limited accuracy in the estimation of global extrema. These inaccuracies depend largely on the selection of the sampling sites and the number of sample points. In the second half of this paper, merit functions are incorporated to each modeling method during the optimization process for the selection of new sample points that lead to an improvement of the current approximation model. The ability of this new procedure to identify global extrema is demonstrated using simple test functions. Nomenclature β constant underlying global portion of Kriging model vector of the unknown coefficients in response surface models b vector of least squares estimators of β C D drag coefficient f constant vector used in Kriging model L sum of the squares of the errors mc merit function k number of design variables n s number of sample points(sites) n t number of test sample points to evaluate mod-eling error r vector of correlation values for Kriging model R(.) correlation function for Kriging model R correlation matrix for Kriging model RM S ub unbiased root mean square error RS response surface x scalar component of x x vector denoting all locations (sites) in the design space x p vector denoting the p th location in the design space X matrix of sample sites for RS model y(. ˆ y(.) estimated model of y(.) vector of correlation parameters for Kriging modeî σ …
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