We propose formulating preliminary-stage aircraft design problems as geometric programs (GPs), which are a specific type of convex optimization problem. Recent advances in convex optimization offer significant advantages over the general nonlinear optimization methods typically used in MDO. Modern GP solvers are extremely fast even on large problems, require no initial guesses or tuning of solver parameters, and guarantee globally optimal solutions. These benefits come at a price: all objective functions and constraints -the mathematical models that describe aircraft design -must be expressed within the restricted functional forms of GP. Perhaps surprisingly, this restricted set of functional forms appears again and again in prevailing physics-based models for aircraft design tradeoffs. Moreover, for various models that cannot be manipulated algebraically into the forms required by GP, we can often fit compact GP models which accurately approximate the original models. The speed of GP solution methods makes their application to large MDO problems a promising approach.

Motivated by practical applications in engineering, this article considers the problem of approximating a set of data with a function that is compatible with geometric programming (GP). Starting with well-established methods for fitting max-affine functions, it is shown that improved fits can be obtained using an extended function class based on the softmax of a set of affine functions. The softmax is generalized in two steps, with the most expressive function class using an implicit representation that allows fitting algorithms to locally tune softness. Each of the proposed function classes is directly compatible with the posynomial constraint forms in geometric programming. Maxmonomial fitting and posynomial fitting are shown to correspond to fitting special cases of the proposed implicit softmax function class. The fitting problem is formulated as a nonlinear least squares regression, solved locally using a Levenberg-Marquardt algorithm. Practical implementation considerations are discussed. The article concludes with numerical examples from aerospace engineering and electrical engineering.

For a UAV to perch on a wire, aircraft control systems which operate far outside typical operating envelopes must be developed. The relevant transient aerodynamics at high angle of attack are not addressed today by control-accessible aerodynamic models. In this work, we present a set of physically-inspired basis functions which have enabled system identification of a nonlinear aerodynamics model along perching trajectories. Data is collected using a motion capture system which, critically, allows free-flight data from real system trajectories to be gathered. When simulated forward, the identified model accurately predicts the observed perching trajectories, making it an indispensable tool for designing feedback controllers that stabilize perching trajectories.

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We present GPkit, a Python toolkit for Geometric and Signomial Programming that prioritizes explainability and incremental complexity. GPkit was designed through an ethnographic approach in the firms, classrooms, and research labs where it became part of the fabric of daily engineering work. Organizations have approached GPkit both in ways which centralize design work and in ways which distribute it, usecases which emerged from and inspired new toolkit features. This twoway flow between mathematical structure and practitioner knowledge resulted in several novel contributions to the formulation and interpretation of convex programs and to our understanding of early-stage engineering design. For example, dual solutions (often considered incidental) can be more valuable to a design process than the "optimal design" itself, and we present novel algorithms and design methods based on this insight.

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This thesis presents a full 1D core+fan flowpath turbofan optimization model, based on first principles, and meant to be used during aircraft conceptual design optimization. The model is formulated as a signomial program, which is a type of optimization problem that can be solved locally using sequential convex optimization. Signomial programs can be solved reliably and efficiently, and are straightforward to integrate with other optimization models in an all-at-once manner. To demonstrate this, the turbofan model is integrated with a simple commercial aircraft sizing model. The turbofan model is validated against the Transport Aircraft System OPTimization turbofan model as well as two Georgia Tech Numerical Propulsion System Simulation turbofan models. Four integrated engine/aircraft parametric studies are performed, including a 2,460 variable multi-mission optimization that solves in 28 seconds.

Due to the coupled nature of aircraft system design, it is important to consider all of the major subsystems when trying to optimize a configuration. This, however, is easier said than done, particularly because each individual subsystem model can be arbitrarily complex, thus making optimization difficult. By restricting an optimization problem to have a certain mathematical structure, significantly more effective and tractable solution techniques can be used. Geometric programming, an example of one such technique, guarantees finding a globally optimal solution. Although it has been shown that geometric programming can be used to solve some conceptual aircraft design problems, the required formulation can prove too restrictive for certain relationships. Signomial programming is a closely related relaxation of geometric programming that offers enhanced expressiveness, but without the guarantee of global optimality. Despite this, solution methods for signomial programs are disciplined and effective. In the present work, signomial programming models are proposed for optimal preliminary sizing of the vertical tail, fuselage, and landing gear of a commercial aircraft with a tubeand-wing configuration. Signomial programming's relaxed formulation allows it to handle some of the key constraints in tail, fuselage, and landing gear design and therefore a significant improvement in fidelity over geometric programming models is achieved. The models are readily extensible and easily combined with other models, making them effective building blocks for a full aircraft model. A primary contribution of this work is to demonstrate signomial programming as a viable tool for multidisciplinary aircraft design optimization.