Date of Award

Winter 1999

Document Type


Degree Name

Doctor of Philosophy (PhD)


Aerospace Engineering

Committee Director

Oktay Baysal

Committee Member

Ponnampalam Balakumar

Committee Member

Jeng-Jong Ro

Committee Member

Manuel D. Salas


A gradient-based shape optimization methodology based on continuous adjoint sensitivities has been developed for two-dimensional steady Euler equations on unstructured meshes and the unsteady transonic small disturbance equation. The continuous adjoint sensitivities of the Helmholtz equation for acoustic applications have also been derived and discussed.

The highlights of the developments for the steady two-dimensional Euler equations are the generalization of the airfoil surface boundary condition of the adjoint system to allow a proper closure of the Lagrangian functional associated with a general cost functional and the results for an inverse problem with density as the prescribed target. Furthermore, it has been demonstrated that a transformation to the natural coordinate system, in conjunction with the reduction of the governing state equations to the control surface, results in sensitivity integrals that are only a function of the tangential derivatives of the state variables. This approach alleviates the need for directional derivative computations with components along the normal to the control surface, which can render erroneous results.

With regard to the unsteady transonic small disturbance equation (UTSD), the continuous adjoint methodology has been successfully extended to unsteady flows. It has been demonstrated that for periodic airfoil oscillations leading to limit-cycle behavior, the Lagrangian functional can be only closed if the time interval of interest spans one or more periods of the flow oscillations after the limit-cycle has been attained. The steady state and limit-cycle sensitivities are then validated by comparing with the brute-force derivatives. The importance of accounting for the flow circulation sensitivity, appearing in the form of a Dirac delta in the wall boundary condition at the trailing edge, has been stressed and demonstrated. Remarkably, the cost of an unsteady adjoint solution is about 0.2 times that of a UTSD solution.

Unlike the Euler equation sensitivities, the Helmholtz equation requires the Hessian of the acoustic field on the control surface. Obtaining accurate Hessian information on curved surfaces is not an easy task, if not impossible. It is been shown that in the natural coordinates, the only required derivative information are the first and second order tangential derivatives of the acoustic field that can be computed very accurately by fitting the boundary variables with a cubic-spline interpolating function. If it were to be attempted, the sensitivities of the Navier-Stokes equations would also require the Hessian of the state variables. Based on above experiences, it is contended that a transformation to the trihedral coordinate system may ease the problem associated with the acquisition of accurate boundary derivative information.