Solving design problems that rely on very complex and computationally expensive calculations using standard optimization methods might not be feasible given design cycle time constraints. Variable fidelity methods address this issue by using lower-fidelity models and a scaling function to approximate the higher-fidelity models in a provably convergent framework. In the past, scaling functions have mainly been either first-order multiplicative or additive corrections. These are being extended to second order. In this investigation variable metric approaches for calculating second-order scaling information are developed. A kriging-based scaling function is introduced to better approximate the high-fidelity response on a more global level. An adaptive hybrid method is also developed in this investigation. The adaptive hybrid method combines the additive and multiplicative approaches so that the designer does not have to determine which is more suitable prior to optimization. The methodologies developed in this research are compared to existing methods using two demonstration problems. The first problem is analytic, whereas the second involves the design of a supercritical high-lift airfoil. The results demonstrate that the krigingbased scaling methods improve computational expense by lowering the number of high-fidelity function calls required for convergence. The results also indicate the hybrid method is both robust and effective. Nomenclature f = objective function g = inequality constraint h = equality constraint l = design space lower bounds u = design space upper bounds W = hybrid weighting value x = design vector β = multiplicative scaling function γ = additive scaling function = trust region size f = objective function convergence tolerance x = design variable convergence tolerance ρ = trust region ratio ∇ = gradient operator ∇ 2 = Hessian operator Subscripts high = high-fidelity model i, j, k = free indices for elements in a vector or matrix low = low-fidelity model m = number of design variables n = current iteration number scaled = scaled low-fidelity value u = unscaled constraint
Using surrogate models in place of high fidelity engineering simulations can help reduce design cycle times and cost by enabling rapid analysis of alternative designs. Surrogate models can also be used in a deliverable product as an efficient replacement for large lookup tables or as a soft sensor to predict quantities than cannot be directly measured. Many different surrogate modeling techniques exist, including new commercial technologies, each with different capabilities and pitfalls. The goal of this research is to aid the designer in selecting the appropriate surrogate model by comparing two popular techniques, second order regression and kriging, along with a new commercial application called Datascape. The three different modeling techniques are compared on model accuracy, computational efficiency, robustness, transparency, and ease of use. The comparisons were done using three test problems: an Earth-Mars transfer orbit problem, the analytic Shekel function, and a low Earth orbit three-satellite constellation design problem. It was found that kriging models performed the best when the sample data used to build the models was sparse, when larger sample sets were used Datascape produced more accurate models.
Due to their current successes, unmanned aerial vehicles (UAVs) are becoming a standard means of collecting information. However, as their missions become more complex and require them to fly farther, UAVs can become large and expensive due to fuel needs. Sidestepping the paradigm of a fixed static wing, the variform concept developed in this paper allows for greater fuel efficiency. Bulky wings could morph into sleeker profiles, reducing drag, as they burn fuel. The development of such wings will rely heavily on computational design exploiting state of the art optimization techniques that account for uncertainty and insure reliability. Nomenclature α Angle of attack η Propeller efficiency ρ ∞ Free stream density c Specific fuel consumption
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