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Why Study Optimization Methods?
The future of aircraft design lies in the tighter integration of the canonical disciplines. The D8 concept proposed tighter integration between aerodynamics and propulsion in the form of Boundary Layer Ingestion could save as much as 15% in fuel burn. The Blended Wing Body and Transonic Truss Braced Wing concepts both trade heavily between structures and aerodynamics to achieve similar predicted gains. The drone revolution has been possible thanks to integration between flight control systems, electric power-trains, and advances in small scale aerodynamics. This tight coupling between disciplines is what necessitated the rise of MDAO in the aircraft design community. However, initial efforts to design aircraft with this increased coupling has come at the cost of a highly complicating the aircraft design space topography.
A few proposals exist to deal with this new and challenging topography. As was mentioned in my thoughts on MDAO, there is a growing body of literature that suggests analysis and optimization must be considered in a unified approach. By carefully constructing the optimization problem, the design space topography can be dramatically simplified, with minimal sacrifices to analysis fidelity. In this approach, optimization methods are integral to the approach by construction, and therefore deserve increased attention.
But let us for a moment be skeptical of these new methods and approach this newly complicated design space from a more conventional angle. As MDAO has become more accepted as standard practice in the aircraft design community, analysis has begun to coalesce around a set of well developed methods, common across all research and professional sectors (see Analysis Models). As a result of the intense focus on analysis methods, optimization takes a back seat and MDAO often turns into simply MDA.
There are a few reasons why optimization tends to be an afterthought in MDAO. First, it is often the case that a true optimum is not sought, and that simply having a design that satisfies the requirements and has reasonable performance is a good enough. The additional time and energy spent moving from a feasible design to an optimal design simply is not worth the trouble. Second, a common adage in aircraft design is that numeric optimization is really good at finding the flaws in your problem formulation, but not so good at actually finding optimal designs. This statement is absolutely true: a numeric optimizer will indeed exploit every weakness of the optimization formulation. In fact, that is exactly what makes numeric optimization successful at finding optima. But in practice, having to chase down every missing constraint can be a huge bother.
Finally, and most critically, designers and subject matter experts frequently don't trust the results a numeric optimization that is viewed as a "black box" beyond their control. I once had a senior engineer tell me a story of an optimizer that had been wrapped around an analysis framework. Due to a trivial error, the optimizer produced an "optimal" aircraft where one of the performance metrics missed the bounds of reasonable by a full order of magnitude. The error was quickly fixed, but the damage had been done. The trust in optimization had been completely lost, and so the whole notion of MDAO was abandoned in favor of a strict MDA only policy.
The case in favor of using optimization is almost self-evident. In an age where we are trying to squeak every edge of performance out of every dollar we spend, optimization is a powerful tool for automating some of the design space search and presenting alternatives to the human the designer. But those of us in academia and research have made an error in attempting to sell optimization as a fully automated process that removes the human designer from the process, and it has cost us the trust of the designers and discipline experts we seek to support. Restoring this faith in a field we know could have tremendous benefit is yet another reason why optimization methods deserve study.
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What Optimization Methods are Currently Used in Aircraft Design?
Two significant fields dominate optimization methods in aircraft design: gradient based algorithms and non-gradient based (or heuristic) methods.
Gradient based algorithms date back to the foundations of calculus and are based on the fundamental principle of a gradient search. A candidate design is selected, the gradient with respect to the design variables is either computed or approximated, and a step is taken in the favorable direction to produce a new candidate design. Among these algorithms are Newton's Method, Quasi-Newton Methods (such as the LBFGS approximation), Sequential Quadratic Programs (SQP), Semi-Definite Programming (SDP), Geometric Programming (GP), Quadratically Constrained Quadratic Programs (QCQP), all of which have received recent attention in either the aircraft design community or the optimization community (and by my estimation may have applications to aircraft design).
Gradient based methods benefit from the ability to mathematically prove that an optimum has been reached via the KKT conditions. However, computing the gradient in a generally complicated multi-disciplinary design space, particularly where black box analyses frequently exist, proves to be a highly difficult task. Methods exist for computing derivatives, such as complex step, automatic differentiation, finite differencing, and adjoint methods (adjoint methods have been demonstrated to be particularly powerful), but no turn-key solution exists to this problem and it remains a significant issue. Additionally, gradient based methods are by construction destined to converge to the closest local optimum unless significant effort is taken to perform a more global search. Whether or not this is a concern has recently been put up for debate, but it remains an open question.
Non-gradient based methods, also called heuristic methods, have recently risen to prominence due to advancements in computing capability. In general, these methods rely on intelligently processing a group of designs through some objective function, and through a time-evolution produce an "optimal" result. Heuristic methods include simplex methods, genetic algorithms (GA), particle-swarm optimization (PSO), and simulated annealing (SA). Each of these methods is utilized across the aircraft design literature.
Heuristics benefit from low setup costs, as no structure is required in the underlying design space and are also highly generalizable to difficult design spaces, such as mixed-integer formulations. However, heuristic methods typically require massive computational resources. Many (such as GA and PSO) are parallelizable, but some (such as simplex and SA) are not. In either case, the runtime on a heuristic method can be excessive. Additionally, heuristics cannot prove optimality and therefore can merely produce candidate designs better than all others considered, as opposed to designs known to be optimal.
Beyond a simple gradient vs. non-gradient deconstruction, one of the most active areas of research is the area of robust optimization and uncertainty quantification. Essentially, this field of optimization seeks to supplement the optimal aircraft design with information about the local area of the design space near the proposed optimum. While great research has been done in this area, it has been an uphill fight to break these methods out of academia due to the complex mathematics, additional overhead, and increased computational cost. In time, these methods (in some form) will most likely become standard practice.
The last area of consideration here is the fledgling field of Machine Learning (ML). Various design efforts have had a great deal of success using ML models for surrogates of more expensive analysis models, but the cost of training data in aircraft design has led to limited expansion beyond this niche. However, these methods continue to produce promising results in other fields and deserve constant evaluation by those of us in the aircraft design community.
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What Research do I Contribute to the Optimization Community?
I am currently developing iterative gradient-based optimization methods which leverage log-convexity to improve convergence rates as compared to existing methods. These methods are pending publication, and will be released with Version 0.0.0 of Corsair.