Inverse problems of determining the unknown shape of oscillating airfoils in compressible 2D unsteady flow via variational technique

A generalized variational principle of 2D unsteady compressible flow around oscillating airfoils is established directly from the governing equations and boundary/initial conditions via the semi-inverse method proposed by He. In this method, an energy integral with an unknown F is used as a trial-functional. The identification of the unknown F is similar to the identification of the Lagrange multiplier. Based on the variational theory with variable domain, a variational principle for the inverse problem (given as the time-averaged pressure over the airfoil contour, while the corresponding airfoil shape is unknown) is constructed, and all the boundary/initial conditions are converted into natural ones, leading to well-posedness and the unique solution of the inverse problems.