AIAA 2001–0922 Using Gradients to Construct Response Surface Models for High-Dimensional Design Optimization Problems

The response surface method (RSM) is one of the most common techniques for building approximation models using values of sampled data alone. In this work, a method for using both function values and their gradients with respect to the design variables in the problem and construct response surface (RS) models has been developed. This approach improves on the efficiency of using the RSM for high-dimensional design optimization problems. Provided that gradient information is available through inexpensive algorithms such as the adjoint method, this approach significantly reduces the large computational cost needed for full RS model construction. The minimum number of function evaluations (CFD analyses, in our case) required for a full quadratic RS model increases with the square of the number of design variables, N , effectively preventing their use in high-dimensional design optimization. In contrast, the proposed modification to the RS model, which incorporates gradient information, can approximate the original function with order N function evaluations, without significantly sacrificing the accuracy of the approximation. After validating the feasibility of the proposed method using simple one-and two-dimensional analytic functions, the approach is applied to the aerodynamic design of a supersonic business jet. The results of this 7-variable design problem indicates that the method can be effectively used to construct response surfaces with greatly reduced computational cost. The advantages of the method become much more significant as the number of design variables increases. Nomenclature α weighting factor controlling the contribution of gradient information for RS construction vector of the unknown coefficients in a response surface model b vector of the least squares estimator of β CD drag coefficient random error L sum of the squares of the errors N number of design variables ns number of sample points (also called sites) nt number of test sample points to evaluate mod-eling error RM S ub unbiased root mean square error RS response surface x scalar component of x x vector denoting locations (sites) in design space X matrix of sample sites for RS model y(.) unknown function ˆ y(.) estimated model function for y(.)