AIAA 2002–5598 Design of a Low-Boom Supersonic Business Jet Using Cokriging Approximation Models

In this paper we study the ability of the Cokriging method to represent functions with multiple local minima and sharp discontinuities for use in the multidimensional design of a low-boom supersonic business jet wing-body-canard configuration. Cokriging approximation models are an extension of the original Kriging method which incorporate secondary information such as the values of the gradients of the function being approximated. Provided that gradient information is available through inexpensive algorithms such as the adjoint method, this approach greatly improves on the accuracy and efficiency of the original Kriging method for high-dimensional design problems. In order to construct Cokriging approximation models, an automated Euler and Navier-Stokes based method, QSP107, has been developed to provide accurate performance and boom data with very rapid turnaround. The resulting approximations are used with a simple gradient-based optimizer to improve a multi-objective cost function with large variations in the design space. Results of sample two-dimensional test problems, together with a 15-dimensional test case are presented and discussed. The Cokriging method is a viable alternative to quadratic response surface methods for preliminary design using a moderate number of design variables, particularly when the cost function being optimized is very nonlinear. Nomenclature β constant underlying global portion of Kriging model CD drag coefficient f constant vector used in Kriging model f c constant vector used in Cokriging model k number of design variables n s number of sample points r vector of correlation values for Kriging model rc vector of correlation values for Cokriging model R(.) correlation function for Kriging model R correlation matrix for Kriging model Rc correlation matrix for Cokriging model x scalar component of x x p vector denoting the p th location in the design space y(.) unknown function ˆ y(.) estimated model of y(.) vector of correlation parameters for Kriging model ˆ σ 2 estimated sample variance