The study of flapping-wing aerodynamics is a problem with very large control space. Adjoint-based approach, by solving an inverse problem, can be used here as an efficient tool for optimization and physical understanding. However, the adjoint equation is typically formulated in a fixed domain. The moving boundary or morphing domain brings in an inconsistency in the definition of arbitrary perturbation at the boundary, which then proposes a new challenge if the control parameters happen to be also at the boundary. An unsteady mapping function, as a usual remedy for such problems, would make the whole formulation too complex to be feasible. Instead, we use non-cylindrical calculus to re-define the perturbation and solve the inconsistency caused by moving/morphing solid boundaries. The approach is first validated for a simple two dimensional test case of a plate plunging in an incoming flow. Then, we apply the approach to reduce the drag of a rigid flapping plate by optimize the phase delay between the plunging and pitching motion as a constant (single parameter) and as a time-varying function (large number of parameters). The extension to three dimensional cases is successfully validated by applying on an oscillatory sphere with incoming flow.
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