Geometric Programming for Aircraft Design Optimization

Abstract

We propose formulating conceptual-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 aircraft design. 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 and constraint functions -the mathematical models that describe aircraft design relations -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 systems. Moreover, we show that 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 and reliability of GP solution methods makes them a promising approach for conceptual-stage aircraft design problems.

Turbofan Engine Sizing and Tradeoff Analysis via Signomial Programming

“…Equation C.26 uses a small angle approximation to set the climb angle, θ. However, the natural logarithm in Equation C.32 is not GP compatible and must be reformulated using the procedure outlined by Hoburg et al [15]. It is not clear apriori how to relax the posynomial equality T SL = T + Lh to an inequality.…”

“…As noted by Martins [21], there exists a need for new multidisciplinary design optimization (MDO) tools that exhibit fast convergence for medium and large scale problems. In pursuit of this goal, Hoburg et al [15] and Kirschen et al [19] have proposed formulating aircraft conceptual design models as geometric programs (GP) or signomial programs (SP). Geometric and signomial programs enable optimization problems with thousands of design variables to be reliably solved on laptop computers in a matter of seconds.…”

Efficient Aircraft Multidisciplinary Design Optimization and Sensitivity Analysis via Signomial Programming

“…Other techniques to make constraints GP compatible include using variable transformations and Taylor approximations. Variable transformations are used in the tail cone model from [5] and the stagnation relations in Appendix B. Taylor approximations are used in the GP-compatible Breguet range equation presented in [2] and the structural model in Appendix A. Additionally, empirical relations can be used to formulate constraints using the methods described in [14]. An example of GP-compatible fits can be found in [5], where XFOIL [15] data is used to generate a posynomial inequality constraint for TASOPT C-series airfoil parasitic drag coefficient as a function of wing thickness, lift coefficient, Reynolds number, and Mach number.…”

“…Equations from [2] for wing structural sizing were adapted using a linearization of the moment and shear load distributions from Appendix C2. The constraints can be applied to both conventional and pi-tails.…”

“…One technique for improving computational efficiency is to solve particular forms of optimization problems rather than general nonlinear programs. Hoburg and Abbeel [2] successfully formulate a basic aircraft design problem as a geometric program (GP), ¶ which is a type of convex optimization problem. GPs with thousands of design variables can be solved on a personal laptop in just a few seconds.…”