A variable-domain variational theory using Clebsch variables for hybrid problems of 2-D transonic rotational flow

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

“…To simplify the computation, it is expedient, without loss of generality, to take the``position vector'' variation r t of the unknown boundaries only in the y-direction, namely (Liu, 1993): r t jy 55…”

“…In the last two decades, with the rapid development of advanced aircraft and turbomachinery as well as the finite element method (FEM), there has been everincreasing interest amongst scientists and engineers in the variational theory of aerodynamics for the inverse/hybrid shape design (Liu, 1989;1993;1996;1998;He, 1999a;1999b;Meauze, 1982). By means of variational theory with variable domain, any unknown boundary (airfoil contour) or discontinuities (shocks and free trailing vortex sheets) can be converted into natural ones, facilitating the finite element solution process (Liu, 1998) and the variational-based meshless (element-free) solution schedule (He, 1999a,b) and, moreover, guaranteeing well-posedness and unique solution of the inverse methods (Liu, 1993). However, almost all inverse design methods available so far are still limited, to the best of the author's knowledge, to steady aerodynamics, possibly due to the fact that the unsteady inverse problem is much more difficult to pose properly ± to formulate as well as to solve.…”