In this paper, we compare and contrast the use of second-order response surface models and kriging models for approximating non-random, deterministic computer analyses. After reviewing the response surface method for constructing polynomial approximations, kriging is presented as an alternative approximation method for the design and analysis of computer experiments. Both methods are applied to the multidisciplinary design of an aerospike nozzle which consists of a computational fluid dynamics model and a finite-element model. Error analysis of the response surface and kriging models is performed along with a graphical comparison of the approximations, and four optimization problems are formulated and solved using both sets of approximation models. The second-order response surface models and kriging models-using a constant underlying global model and a Gaussian correlation function-yield comparable results.

Response surface methods have been used for a variety of applications in aerospace engineering, particularly in multidisciplinary design optimization. We investigate the use of kriging models as alternatives to traditional second-order polynomial response surfaces for constructing global approximations for use in a real aerospace engineering application, namely, the design of an aerospike nozzle. Our objective is to examine the dif culties in building and using kriging models to create accurate global approximations to facilitate multidisciplinary design optimization. Error analysis of the response surface and kriging models is performed, along with a graphical comparison of the approximations. Four optimization problems are also formulated and solved using both sets of approximation models to gain insight into their use for multidisciplinary design optimization. We nd that the kriging models, which use only a constant "global" model and a Gaussian correlation function, yield global approximations that are slightly more accurate than the response surface models.

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As part of phase 2 of the X-33 Program, NASA selected an integrated lifting body/aerospike engine con guration as the study vehicle for the conceptual analysis of a single-stage-to-orbit reusable launch vehicle. A team at NASA Langley Research Center participated in the screening and evaluation of a number of proposed vehicle con gurations in the early phases of the conceptual design process. The performance analyses that supported these studies were conducted to assess the effect of the vehicle's lifting capability, linear aerospike engine, and metallic thermal protection system on the weight and performance of the vehicle. These performance studies were conducted in a multidisciplinary fashion that indirectly linked the trajectory optimization with weight estimation and aerothermal analysis tools. This approach was necessary to develop optimized ascent and entry trajectories that met all vehicle design constraints. Signi cant improvements in ascent performance were achieved when the vehicle ew a lifting trajectory and varied the engine mixture ratio during ight. Also, a considerable reduction in empty weight was possible by adjusting the total oxidizer-to-fuel and liftoff thrust-to-weight ratios. However, the optimal ascent ight pro le had to be altered to ensure that the vehicle could be trimmed in pitch using only the ow diverting capability of the aerospike engine. Likewise, the optimal entry trajectory had to be tailored to meet thermal protection system heating rate and transition constraints while satisfying a crossrange requirement. NomenclatureC L = lift coef cient I sp = speci c impulse, s M e = edge Mach number M 1 = freestream Mach number O=F = total oxidizer-to-fuel ratio q = dynamic pressure, psf q ¢ ® = dynamic pressure times angle-of-attack,psf ¢ deg Re µ = momentum thickness Reynolds number S = aerodynamic reference area, ft 2 T =W = thrust-to-weightratio .T =W / eng = engine thrust-to-weightratio W = entry weight, lb W empty = empty weight, lb W ins = inserted weight, lb X=L = body position over vehicle length ® = angle of attack, deg 1 payload = change in payload from

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