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This paper presents an adjoint method for the multi-objective aerodynamic shape optimization of unsteady viscous flows. The goal is to introduce a Mach number variation into the Non-Linear Frequency Domain (NLFD) method and implement a novel approach to present a time-varying cost function through a multi-objective adjoint boundary condition. The paper presents the complete formulation of the time dependent optimal design problem. The approach is firstly demonstrated for the redesign of a helicopter rotor blade in two-dimensional flow and in three-dimensional viscous flow, the technique is employed to validate and redesign the NASA Rectangular Supercritical Wing (RSW).

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