Global Optimization of Wing Aerodynamic Optimization Case Exhibiting Multimodality

Abstract

An investigation into an aerodynamic optimization benchmark case that exhibits multimodality is presented using a global optimization approach. The recently suggested case 6 of the AIAA Aerodynamic Design Optimization Discussion Group involves the drag minimization of a rectangular NACA0012 wing subject to lift and root bending moment, as well as other geometric constraints. In this paper, optimization of that case using a state-of-the-art constrained population-based global optimization framework is presented.Shape control is achieved via a hierarchical application of the radial basis function domain element approach. Three different fidelity of optimization cases are performed to investigate the multimodality of various aspects of the full problem, with four independent runs of each case being performed. When considering chord optimization, the optimizer converges to two independently identifiable minima with very similar performance. When adding dihedral and sweep variation, further multimodality is introduced, though this multimodality is reasonably well defined, with minima including forward and rearward sweep, and an upward and downward winglet. Introducing thickness variables leads to a highly multimodal problem due to the coupling between the chord and thickness variables. The theoretical minimum drag for this case is 24.7 counts, which a number of the runs get close to achieving.

Identifying Multiple Optima in Aerodynamic Design Spaces

“…As previously covered in Poole et al [37], the root bending moment constraint of this problem inhibits the conventional optimal elliptic loading result. Instead, the optimal loading of this result is given by the solution of Jones [54] (which is a specific solution of a result later proved by Klein and Viswanathan [55]):…”

“…where δ = 8( − 1) 2 , hence this varies with aspect ratio. Poole et al [37] showed that the theoretical minimum for this problem occurs at the maximum allowable span, and is 24.7 counts. However, while Case 6 is interesting, it permits a substantial degree of geometric flexibility, hence it is useful to consider a slightly simpler problem.…”

“…For aerofoil optimization, it has been suggested that ASO problems exist that are unimodal [30] and multimodal [31,32], while the same is true of wings exhibiting unimodality [33] and multimodality [30]. Recently, results of the optimization of a rectangular wing in inviscid subcritical flow (this is case 6 of the AIAA ADODG [34]) have been presented investigating the degree of multimodality in that case [35][36][37][38]. However, while some aerodynamic optimization problems are multimodal, there is little effort to attempt to identify and characterise these.…”

“…[36], and Streuber and Zingg [38]. The current authors performed similar work by running a heuristic global optimizer multiple times on the same problem to analyse whether it converged to different solutions each time [37].…”

Constrained niching using differential evolution

“…Despite the lack of theoretical research into constrained multimodal optimization, there exists an appetite for such methods, particularly in engineering optimization. For example, a recently-proposed shape optimization problem requires finding all the optima of an aircraft wing optimization problem subject to a substantial number of linear and non-linear constraints [29]. Furthermore, other constrained engineering optimization problems exhibit a strong multimodal tendency, for example in truss design [30], and as such, studies and methods for solving such problems are required.…”