Date of Award

Spring 1997

Document Type


Degree Name

Doctor of Philosophy (PhD)


Aerospace Engineering

Committee Director

Oktay Baysal

Committee Member

Colin P. Britcher

Committee Member

Osama Kandil

Committee Member

David E. Keyes


A gradient-based shape optimization methodology based on quasi-analytical sensitivities has been developed for practical three-dimensional aerodynamic applications. The flow analysis has been rendered by a fully implicit, finite-volume formulation of the Euler and Thin Layer Navier-Stokes (TLNS) equations. The flow equations and aerodynamic sensitivity equation have been solved using an alternating-direction-implicit (ADI) algorithm for memory efficiency. A wing geometry model based on space-surface and planform parameterization has been utilized. The present methodology and its components have been tested via several comparisons.

Initially, the inviscid flow analysis for a wing has been compared with those obtained using an unfactored, Preconditioned Conjugate Gradient (PCG) approach, and an independent Computational Fluid Dynamics (CFD) code which has been extensively validated. Then, the viscous laminar flow analysis for a wing has been compared with that obtained using again the extensively validated CFD code. Next, the sensitivities computed with the present method have been compared with those obtained using the finite-difference and the PCG approaches. Effects of convergence tolerance on the flowfield sensitivities have been shown. Also, effects of grid size and viscosity on the flow analysis, sensitivity analysis and the shape optimization have been established.

Despite the expected increase in the computational time, the results indicate that shape optimization problems, which require large numbers of grid points, can be resolved with a gradient-based approach.

The new procedure has been demonstrated in the design of a cranked arrow wing at Mach 2.4, with coarse and fine grid based computations performed with Euler and TLNS equations. The influence of the initial constraints on the geometry and aerodynamics of the optimized shape has been explored. Various final shapes generated for an identical initial problem formulation but with different optimization path options (coarse or fine grid, Euler or TLNS) have been aerodynamically evaluated via a common fine grid TLNS based analysis. The efficacy of these design options has been evaluated by comparing net performance improvement in tandem with the CPU time requirements.

Results show that fluid dynamic and sensitivity analyses using ADI compare well with the PCG method and CFL3D code. The ADI method reduces the memory storage but increases the computing time as compared to the PCG method. It is demonstrated that the inherent larger size of optimization problems can be accommodated by using the ADI method. The presently developed optimization procedure is capable of learning aerodynamic lessons during the evolution of optimized shapes. Initial constraints conditions show significant bearing on the optimization results. Results demonstrate that to produce an aerodynamically efficient design, it is imperative to include viscous physics in the optimization procedure with proper resolution. However, if CPU time constraints do not permit this option, it is advantageous to incorporate inadequately resolved viscous flow physics in lieu of properly resolved inviscid flow physics.

Based upon the present results, it is recommended that to better utilize computational resources, a number of viscous coarse grid cases using the PCG (preferably) or ADI method, should initially be explored to improve optimization problem definition, design space and initial shape. Optimized shapes should be analyzed using high fidelity (viscous fine grid resolution) flow analysis to evaluate their true performance potential. Subsequently, a viscous fine grid-based shape optimization should be conducted, using ADI method, to accurately obtain the final optimized shape.