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There is a need to adapt and improve conceptual design methods through better optimization, in order to address the challenge of designing future engineered systems. Aerospace design problems are tightly-coupled optimization problems, and require all-at-once solution methods for design consensus and global optimality. Although the literature on design optimization has been growing, it has generally focused on the use of gradient-based and heuristic methods, which are limited to local and low-dimensional optimization respectively. There are significant benefits to leveraging structured mathematical optimization instead. Mathematical optimization provides guarantees of solution quality, and is fast, scalable, and compatible with using physics-based models in design. More importantly perhaps, there has been a wave of research in optimization and machine learning that provides new opportunities to improve the engineering design process. This thesis capitalizes on two such opportunities.
The first opportunity is to enable efficient all-at-once optimization over constraints and objectives that use arbitrary mathematical primitives. This work proposes a constraint sampling and learning approach for global optimization, leveraging developments in machine learning and mixed-integer optimization. More specifically, the feasible space of intractable constraints is sampled using existing and novel design of experiments methods, and learned using optimal classification trees with hyperplanes (OCT-Hs). OCT-Hs describe union-of-polyhedra approximations of intractable constraints, which are solved efficiently using commercial solvers to find near-feasible and near-optimal solutions to the global optimization problem. The constraints are then checked and the solution is repaired using projected gradient methods, ensuring feasibility and local optimality. The method is first tested on synthetic examples, where it finds the global optima for 9 out of 11 benchmarks, and high-performing solutions otherwise. Then it is applied to two real-world problems from the aerospace literature, and especially to a satellite on-orbit servicing problem that cannot be addressed via other global optimization methods. These applications demonstrate that decision tree driven optimization provides efficient, practical and optimal solutions to difficult global optimization problems present in aerospace design as well as other domains, regardless of the form of the underlying constraints.
The second opportunity is to optimize designs affected by parametric uncertainty in a tractable and deterministic manner, while providing guarantees of constraint satisfaction. Inspired by the wealth of literature on robust optimization, and specifically on robust geometric programming, this thesis proposes and implements robust signomial programming to solve engineering design problems under uncertainty. The methods are tested on a conceptual aircraft design problem, demonstrating that robust signomial programs are sufficiently general to address engineering design problems, solved efficiently by commercial solvers, and result in designs that protect deterministically against uncertain parameter outcomes from predefined sets. In addition, robust designs are found to be less conservative than designs with margins; robust aircraft demonstrate 9% better average performance than aircraft designed with margins over the same scenarios, while providing guarantees of constraint feasibility.
In anticipation of future aerospace design problems becoming increasingly coupled, complex and risky, this thesis provides a new perspective for dealing with design challenges using structured mathematical optimization. The proposed methods inject mathematical rigor into engineering design methods while keeping practical concerns for conceptual design in focus. |
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