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High Fidelity Numerical Optimization Using Unstructured-Grid Methods

An efficient design method for detailed aerodynamic shape optimization for safety and performance of space vehicles. Fully automated geometry changes allow for optimization of complete vehicles including trimming surface deflections at single or multiple design points.

This technology represents a state-of-the-art tool for design that will be critical to the development of exploration vehicles. The capability was used extensively to support the Space Launch Initiative Programs, both in the application of a high fidelity capability for improved Industry conceptual design, and to support in-house concept design and refinement in support of technology maturation studies.

examples of surface mesh

Right: Examples of surface mesh perturbations available for optimization on the Ames CTV.

Research Overview
Since the days of the High Speed Research (HSR) program, the unstructured-grid Euler computational fluid dynamics (CFD) algorithm, AIRPLANE, has been coupled to a constrained gradient optimization method. In order to facilitate optimization, a suite of tools has been developed to support optimization, and the computational efficiencies of several of the tools have been improved. In addition to the flow solver and optimization method, the optimization package includes: a surface mesh perturbation method, APSHAPER; a volume mesh perturbation method, MESHMV; the unstructured mesh generator, MESH3D, and an adjoint solver, SYNPLANE (commercially available software).

The triangulated surfaces are perturbed during optimization using APSHAPER. The design variables are factors applied to user-defined sets of analytic shape functions added to the configurations’ surfaces. The optimizer adjusts these to improve an objective function that is typically based on an aerodynamic quantity.

The designer is provided with camber and thickness design variables, polynomial functions for leading- and trailing-edge droop, exponential functions to modify the leading-edge bluntness, twist, and longitudinal and vertical displacement functions to modify the planform or dihedral of the wing. Flaps (control surfaces) can be modeled with arbitrary hinge lines with or without spanwise surface gaps. The design variables allow for geometry changes on wing, body, and tail surfaces which can permit geometric changes that cover the entire surface of a configuration. In addition, this method allows for individual component surfaces to be translated or perturbed and then merged together automatically during the course of optimization. An example of this is a wing moving upwards or downwards on a stationary fuselage.

Pitching moment

Left: Pitching moment coefficients for baseline and optimized Apollo CM.

Volume grids of tetrahedra are generated with MESH3D for the baseline configuration, then these volume grids are perturbed rather than re-generated, if possible, via MESHMV. SYNPLANE is an adjoint method that provides an alternative way to compute gradients which significantly improves the computational costs of optimization. Its origin is from the mathematical theory of optimal control, and a similar structured-grid based method was used extensively in the HSR program.

Several vehicles have been optimized via the unstructured grid approach; these include: Orbital Space Planes, the Ames Crew Transfer Vehicle, and more recently the Apollo Command Module (CM). The results of multipoint optimization on the OSP and CTV resulted in improved performance and stability of the vehicle at multiple design points. The CTV configuration’s performance and handling qualities were verified with a wind tunnel test, Navier-Stokes computations, and flight simulation in the Ames Vertical Motion Simulator.

The baseline Apollo CM (employed on all flights) is stable and trimmed in a dangerous apex forward position which poses a safety risk if the CM separates from the launch tower during abort. Optimization was employed on the Apollo CM to remedy this undesirable characteristic. The design variables included: apex radius (RA), base radius (RO), corner radius (RC), and the included angle, q.

The CM design exercise indicated that cross-range performance can be improved while maintaining robust apex-aft stability with a single trim point. The pitching moment characteristics obtained with AIRPLANE are shown for the baseline and optimized CM, and a Navier-Stokes solution of the optimized configuration confirms the Euler results. The optimization was performed at three design conditions at Mach 3.3 for angles of attack of 75, 130, and 143 degrees. The design goals were to force a positive pitching moment at the first design point (a =75), to eliminate the apex forward trim point, to improve the L/D at a =130, and to maintain the base forward trim point at a =143. The CM design exercise indicated that cross-range performance can be improved while maintaining robust apex-aft stability with a single trim point. The optimized CM has a larger base radius, a larger included cone angle, and a smaller corner radius than the Apollo CM.

computation of optimized Apollo

Right: Navier-Stokes computation of the optimized Apollo capsule shape with surface streamlines and pressure contours shown.

Aerodynamic shape optimization using structured grids was used successfully for several years at NASA Ames on High Speed Civil Transport designs. An adjoint method enabled rapid gradient computation, and efficient surface and volume mesh movement techniques were applied. However, these structured grid methods required start-up periods of a few months of labor to generate suitable meshes for complete configurations.

Unstructured grid methods can establish the baseline surface and volume grid in far less time and more easily than for structured grid methods. Volume grid generation is completely automated independent of the geometric complexity of the configuration. Complete configuration multipoint numerical optimization can begin within a few days using unstructured methods, and optimization results can be obtained within a couple of weeks with a dedicated computer system, potentially before a structured-grid based optimization program could even begin.