Response surface estimation and refinement in collaborative optimization

Abstract

This paper describes the use of response surface estimation in collaborative optimization, an architecture for large-scale multidisciplinary design. Response surfaces are used to model the disciplinary subproblem optimization results as a function of the interdisciplinary target variables. These models are then used to find a new design point, about which new disciplinary response surfaces are constructed. The utility of response surface estimation in collaborative optimization depends upon the generation of inexpensive accurate response surface models and the refinement of these models over several fitting cycles. Special properties of the subproblem optimization formulation are exploited to reduce the number of required subproblem optimizations to develop a quadratic model from O(n) to O(n). Implicit information is shown to reduce the number of required suboptimizations by a further 50% in some cases. Response surface refinement is performed using ideas from trust region methods. Results for the combined approaches are demonstrated and compared on two problems: a 44 variable tailless UAV example, using relatively simple analyses, and a substantially more complex HSCT design, with finite element structural analysis and a higher order aerodynamic model. Nomenclature a = maneuver angle of attack b = span B = coefficients in response surface fit C = coefficients in response surface fit Cj = local constraints Cm = pitching moment coefficient C[ = section lift coefficient 5e = eleven deflection angle Tl = propeller efficiency frs = response surface approximation of f m = number of unknown coefficients in RS fit n = number of targets to CO subproblem N = number of CO subproblems O()= order of R = computed aircraft range R0 = target aircraft range sfc = specific fuel consumption tj = spar cap thicknesses 0j = section jig-twist Wj = penalty weight W = weight fraction local design variable W0 = target weight fraction (winilial/wfmal) Wr t= trim ballast weight, tip and root Xj = local design variable; those with a corresponding system level specified target, x = strictly local design variables; those without corresponding system level specified targets yi = local computed state variable, a function of x or x, with a corresponding target Zj. a = maneuver angle of attack Zj = system level targets