Adjoint-based shape and kinematics optimization of flapping wing propulsive efficiency

Abstract

Optimization of the 3-D unsteady viscous flow near a flapping wing is performed using a time-dependent adjoint-based methodology developed in [AIAA 2008-5857 and AIAA J. Vol.48, No.6, pp.1195-1206, 2010]. Sensitivities of the thrust and propulsive efficiency to wing shape and kinematic parameters are computed using the time-dependent discrete adjoint formulation. The unsteady discrete adjoint equations required for calculation of the sensitivity derivatives are integrated backward in time over the entire interval of interest. The gradient of the objective functional obtained using the adjoint formulation is then used to update the values of shape and kinematic design variables. The efficiency of this adjoint-based methodology is demonstrated by optimizing shape and kinematics of a wing undergoing insect-based flapping motion. Our numerical results show that the highest improvement in the thrust and propulsive efficiency is obtained by using the combined optimization of wing shape and kinematics. I. Introduction Insects and small birds represent fully functional examples of efficient small-scale flying devices. However, copying of wing kinematics and shape of flying animals is far from being sufficient to design and build effective, highly maneuverable, agile micro air vehicles (MAVs). Indeed, the current state-of-the-art materials, micro-scale actuators, propulsion systems, and power sources are different and in most cases less efficient than those created by Mother Nature over millions years of evolution. This lack in efficiency of currently available MAV components indicates that a different region of the design space than that associated with flying insects and animals should be explored to be able to maximize the performance of flapping-wing microsystems. Therefore, designs inspired by flying animals can be used only as a preliminary conceptual design that requires further optimization for constructing efficient and agile flying micro-scale platforms optimized for size, weight, speed, and maneuverability. This is a very challenging optimization problem that involves hundreds or even thousands kinematics and shape design variables and is governed by highly unsteady vortex-dominated turbulent flows. Therefore, efficient, mathematically rigorous optimization techniques based on optimal control theory should be used for solving this class of problems. In spite of significant progress in modeling and computational fluid dynamics (CFD) analysis of flappingand rotary-wing platforms [1-5], questions related to optimal design of efficient micro air vehicles (MAV) have not yet been properly addressed especially in three dimensions because of the complicated physical phenomena and computational cost involved. Various parametric and sensitivity studies (e.g., see [1]) have revealed that there is an essentially nonlinear relationship between the major wing kinematic parameters (amplitude, frequency, phase shift angle), shape parameters (wing planform, twist, and thickness), and global flow parameters (the Reynolds, Strouhal, and Mach numbers). Conventional parametric studies, which estimate the sensitivity to each individual design variable independently, do not take into account this nonlinear relationship between the main parameters determining the MAV performance. Furthermore, parametric studies are extremely computationally expensive because of the very large dimensionality of the design space and therefore impractical for optimization and design of efficient flapping-wing microsystems. Several attempts have recently been made to use genetic algorithms based on low-fidelity models [6], highfidelity models [7], and experimental apparatus [8] for optimization of flapping-wing flows. Since these stochastic optimization techniques require thousands of evaluations of the objective functional and consequently thousands of solves of the unsteady flow equations for each design variable, all these approaches have been limited to optimization of 2-D flows with a very small number (less than 4) of design variables. Gradient-based methods provide a powerful alternative for optimization of flapping airfoils and wings. Culbreth et al. [9] uses a finite difference method coupled with a 3-D Navier-Stokes solver to evaluate the