Continuous Adjoint Sensitivities for Optimization with General Cost Functionals on Unstructured Meshes

A continuous adjoint approach is developed to obtain the sensitivity derivatives for the Euler equations. The completederivationofthecostateequations andtheirtransversality (boundary)conditionsarepresented. Both the state and the costate equations are second-order e nite volume discretized for unstructured meshes, and they are coupled with a constrained optimization algorithm. Also integrated into the overall methodology are a geometry parameterization method for the shape optimization, and a dynamic unstructured mesh method for the shape evolution and the consequent volume mesh adaptations. For the proof of concept, three transonic airfoil optimization problems are presented. This method accepts general cost functionals, which are not necessarily functions of pressure only. It is also shown that a switch to the natural coordinate system in conjunction with the reduction of the governing state equation to the control surface results in sensitivity integrals that are only a function of the tangential derivativesof the state variables. Thisapproach eliminates the need fornormal derivativecomputations that can be erroneous.