Analytical Sensitivity Analysis of an Unsteady Vortex Lattice Method for Flapping Wing Optimization

Abstract

This work considers the design optimization of a flapping wing in forward flight with active shape morphing, aimed at maximizing propulsive efficiency under lift and thrust constraints. This is done with an inviscid three-dimensional unsteady vortex lattice method, whose lack of fidelity is offset by a relatively inexpensive computational cost. The design is performed with a gradient-based optimization, where gradients are computed with an analytical sensitivity analysis. Wake terms provide the only connection between the forces generated at disparate time steps, and must be included to compute the derivative of the aerodynamic state at a time step with respect to the wing shape at all previous steps. The cyclic wing morphing, superimposed upon the flapping motions, is defined by a series of spatial and temporal approximations. The generalized coordinates of a finite number of twisting and bending modes are approximated by cubic splines. The amplitudes at the control points provide design variables; increasing the number of variables (providing the wing morphing with a greater degree of spatial and temporal freedom) is seen to provide increasingly superior designs, with little increase in computational cost. I. Introduction HE design and optimization of artificial flapping wing flyers presents considerable difficulties in terms of computational cost: the complex physical phenomena associated with the flight (unsteady low Reynolds number vortical flows in conjunction with a nonlinear elastic wing surface undergoing large prescribed rotations and translations) may require a high-fidelity computational tool. Furthermore, the search optimization process typically requires many function evaluations to converge to a relevant optimum. Lower fidelity numerical tools may help alleviate the burden, either used during the search process in conjunction with a higher-fidelity model 1