Three-dimensional aerodynamic shape sensitivity analysis and design optimization using the Euler equations on unstructured grids

Developed is an approach whereby aerodynamic shape sensitivity analysis and design optimization are pedormed on three-dimensional unstructured meshes. The advantage of unstructured grids (when compared with a structured-grid approach) is their inherent ability to discretize irregularly shaped domains with greater eficiency and less effort. Hence, this approach is ideally suited for geometrically complex configurations of practical interest. The nonlinear Euler equations are solved using a fullyimplicit, upwind, cell-centered, finite-volume scheme. The discrete, linearized systems which result from this scheme are solved iteratively by a preconditioned conjugate-gradient-like algorithm known as GMRES; a similar procedure is also used to solve the accompanying linear aerodynamic sensitivity equalions in incremental iterative form. As shown, this particular form of the sensitivity equations makes large-scale gradient-based aerodynamic optimization possible by taking advantage of memory eSJicient methods to construct exact matrix-vector products. Wing-planform parameterization is trccornplished via scaling and translation factors at pre-selected locations along the wing span, then linearly varying these factors between locations. Once the surface has been deformed, the unstructured grid is adapted by considering the mesh as a system of interconnected springs. Grid sensitivities are obtained by differentiating the surface parameterization and the grid adaptation algorithms with ADIFOR (which is an advanced automatic-diferentiation software tool). To evaluate this shape optimization procedure, the planform shape of an initially rectangular wing with uniform NACA-0012 cross-sections is optimized in a compressible, inviscid flow. 1. Introduct ion As recently noted by Reuther et al. [ 11 “while flow analysis has maturecl to the extent that Navier-Stokes calculations are routinely carried out over very complex configurations, direct CFD based design is only just beginning to be used in the treatment of moderately complex three-dimensional coizfgurations”. This is primarily due to the fact that to generate a single structured grid about such a configuration is difficult, if not impossible. Thus, to handle geometry of practical interest, some sort of domain decomposition scheme must be incorporated into the design code. For structured grid solvers, these techniques would include multiblocked, zonally patched, and overlapped (sometimes referred to as Chimera) grid algorithms. However, as the geometric flexibility of the method increases, so does the complexity of the underlying algorithm. Since the use of sensitivity analysis, to evaluate the needed gradients for a numerical optimizer, is still evolving, little work has been done toward extending these algorithms to include these domain decomposition methods. The research which has been accomplished has mostly concentrated on the use of niultiblocked grids. On this, Reuther et al. [ I ] have developed a multiblock-multigrid adjoint solver (“variational” or “control theory” approach [ 2 ] ) which was applied for the wing redesign of a transonic business jet. Eleshaky and Baysal [3] developed a multiblock “discrete” adjoint solver which was applied to a simple axisynlmetric nozzle near a flat plat. As for the use of the more advanced domain decomposition methods (zonal and overlapped grids), and combinations of the three various types, Taylor [4] has differentiated an advanced flow-analysis code to perform the discrete sensitivity analysis. * Graduate Research Assistant. Student Member, AIM # Associate Professor. Member, AIAA. Copyright