A Systematic Formulation of the Continuous Adjoint Method Applied to Viscous Aerodynamic Design

A continuous adjoint approach to aerodynamic design for viscous compressible flows on unstructured grids is developed, and three important problems raised in the continuous adjoint literature are solved: using tools of shape deformation of boundary integrals a generic adjoint formulation is developed with independence of the kind of mesh used; a systematic way of reducing the 2 order derivative terms which arise is presented which avoids the need of using higher order numerical solvers to obtain accurate approximations of the 2 order derivatives; and finally, the class of admissible optimization functionals is clarified. Several remarks are made concerning the longstanding discrete vs. continuous adjoint dichotomy, with the emphasis not on the advantages or disadvantages of each method, but rather on the well-posedness of the approaches. The accuracy of the sensitivity derivatives is assessed by comparison with finite-difference computations, and the validity of the overall methodology is illustrated with design examples under demanding subsonic conditions.