Energetically Optimal Flapping Wing Motions via Adjoint-Based Optimization and High-Order Discretizations

Abstract

A globally high-order numerical discretization of time-dependent conservation laws on deforming domains, and the corresponding fully discrete adjoint method, is reviewed and applied to determine energetically optimal flapping wing motions subject to aerodynamic constraints using a reduced space PDE-constrained optimization framework. The conservation law on a deforming domain is transformed to one on a fixed domain and discretized in space using a high-order discontinuous Galerkin method. An efficient, high-order temporal discretization is achieved using diagonally implicit Runge-Kutta schemes. Quantities of interest, such as the total energy required to complete a flapping cycle and the integrated forces produced on the wing, are discretized in a solver-consistent way, that is, via the same spatio-temporal discretization used for the conservation law. The fully discrete adjoint method is used to compute discretely consistent gradients of the quantities of interest and passed to a black-box, gradient-based nonlinear optimization solver. This framework successfully determines an energetically optimal flapping trajectory such that the net thrust of the wing is zero to within 9 digits after only 12 optimization iterations.