Convex shape optimization of aerospace vehicles

The field of aerodynamic shape optimization (ASO) has seen dramatic advances in the last several decades through major developments in computational power and numerical methods. Today, the process of gradient-based optimization of complex, three-dimensional aerospace vehicles, using high-fidelity physical models, is mature. This dissertation explores a new horizon of shape optimization for aerospace vehicles, employing techniques from the field of convex optimization. Unlike traditional nonconvex, gradient-based optimization techniques, which iteratively refine an initial design towards a new local optimum, convex optimization seeks a globally optimal solution. These techniques address two important open problems in ASO. The first is the need to be able to efficiently and robustly explore high dimensional, constrained, multi-objective design spaces in order to assess performance tradeoffs and limits during the initial development of new vehicles. The second problem is linking these exploratory, or conceptual design, studies to the high-fidelity, gradient-based optimization frameworks for final refinement. These frameworks require a parameterization and an initial design point. Good parameterizations and initial design points can significantly decrease the overhead of these computationally expensive processes. Convex optimization offers an exciting avenue to addressing both of these problems. In the first part of this dissertation, I develop a framework called Convexity Assisted Shape Optimization, or CASO. CASO provides a set of rules and requirements for approaching aerospace vehicle shape optimization problems through the lens of convex optimization. I also propose two new types of smooth and accurate convex surrogates that will be useful in reducing this framework to practice. In the second part of this dissertation, I propose several new classes of orthogonal basis functions for parameterizing shapes in aerodynamic shape optimization problems. In some cases, these bases simplify the derivation and expression of useful aerodynamic objective functions. In other cases they offer a natural path to representing important aspects of aerodynamic shapes. I also show how these bases can be used to develop convex formulations of several common aerodynamic performance indicators, spanning multiple flow regimes. In the third part, I extend these methods to nonconvex objective functions that have convex trust regions that may be represented accurately and smoothly using convex surrogates. I also consider the cases of nonconvex objective functions that benefit from a transformation and relaxation strategy or a bi-level optimization scheme to preserve the ability to identify a global optimum. Finally, I show how these methods can be applied to actual design problems and link these conceptual results to a high-fidelity design framework. These design problems span multiple flight regimes, performance indicators, and shape representations, in order to provide a broad sampling of the types of problems that can be approached using CASO.

Keywords: Engineering.
Aeronautics and Astronautics.
Advisor: Alonso Juan José
Kochenderfer Mykel J.
Lall Sanjay
Schwager Mac