A variable-domain variational formulation of inverse problemIA of 2-D unsteady transonic flow around oscillating airfoils

The present paper carries out, for the first time, a detailed theoretical investigation on the inverse problem in unsteady aerodynamics. Special attention is paid to finding proper ways of problem-posing and mathematical formulation. To demonstrate the basic idea, only an inverse problem of typeIA of unsteady transonic flow with shocks around oscillating airfoils is studied herein. It has been formulated by a family of variational principles (VP) with variable domain, in which all unknown boundary (airfoil contour) and discontinuities (shocks and free trailing vortex sheets) are handled (captured) via the functional variation with variable domain. As a result, almost all boundary- and interface-conditions have been converted into natural ones. Thus, a rigorous theoretical basis for unsteady airfoil design and finite element (FE) applications is provided. On the basis of these variational principles developed in this paper, a method using new self-deforming finite element is suggested for the numerical realization of the variable-domain variation of the functional and a numerical example is given. Its suitability and effectiveness are demonstrated by the numerical results.