By Andras Sobester, Alexander I J Forrester
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There may come a point when aerodynamicists will feel the need for a comparable amount of shape- and topological-freedom, but, for now, a surface lofted over 2D sections is the status quo of the industry, and it is the philosophy adopted in this book too. We exemplified the above idea in this chapter through the problem of fuselage modelling. We showed that we can start with 2D sketches and move into the third dimension by introducing a series of functions of a streamwise variable (x is an obvious choice) that describe the shape and positioning of the parametric cross-sections, over which we can then loft a surface (in fact, if these streamwise functions yield a value for any X, they define the ‘loft’ too), as well as a profile section that determines the scaling and positioning of the support curves of the loft.
6 The basin of attraction of a local minimum is the geometrical locus of all those potential gradient-descent starting points from which the optimizer will go to that local minimum. 7 Certainly not at macro-scales – there may be an argument for trying to evolve complex shapes for, say, very small features designed to control boundary-layer behaviour. 4 A Parametric Fuselage: A Case Study in the Trade-Offs of Geometry Optimization The greatest single challenge of the above wish-list is that improvements on one count can usually only be made at the expense of another.
As a result, a large proportion of aircraft with pressurized cabins feature a fuselage with a circular cross-section. 15. With the shape thus settled, the only other element we need for this optimization exercise is a uniform scaling variable. Technically, we also need to vary the vertical position of the scaled (semi-)circle placed around the constraint points, but this problem can, to some extent, be separated from the main optimization problem, so we could refer to the latter as a one-variable problem, with the proviso that each tentative solution has to undergo a simple additional search (a kind of repair step) to find its optimum position (this will be the case for the next two stages of the search too).