Third-order sensitivity analysis for robust aerodynamic design using continuous adjoint

Robust design problems in aerodynamics are associated with the design variables, which control the shape of an aerodynamic body, and also with the so-called environmental variables, which account for uncertainties. In this kind of problems, the set of design variables, which leads to optimal performance, taking into account possible variations in the environmental variables, is sought. One of the possible ways to solve this problem is by means of the second-order second-moment approach, which requires first-order and second-order derivatives of the objective function with respect to the environmental variables. Should the minimization problem be solved using a gradient-based method, algorithms for the computation of up to third-order sensitivity derivatives (twice with respect to the environmental variables and once with respect to the shape controlling design variables) must be devised. In this paper, a combination of the continuous adjoint variable method and direct differentiation to compute the third-order sensitivities is proposed. This is shown to be the most efficient among all alternative methods provided that the environmental variables are much less than the design ones. Apart from presenting the method formulation, this paper focuses on the assessment of the so-computed up-to third-order mixed derivatives through comparison with costly finite-difference schemes. To this end, the robust design of a two-dimensional duct is performed. Then, using the validated method, the robust design of a two-dimensional cascade airfoil is demonstrated. Although both cases are handled as inverse design problems, the method can be extended to other objective functions or three-dimensional problems in a straightforward manner. Copyright (c) 2012 John Wiley & Sons, Ltd.