By Andras Sobester, Alexander I J Forrester
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Additional info for Aircraft Aerodynamic. Design Geometry and Optimization
Conversely, specifying, say, only its three angles would leave the triangle underparameterized, as an infinity of triangles exist with the same three angles. Such parametric formulations are said to be ill-posed. Incidentally, the latter is also an example of the separation of shape and scale, as defining the angles of the triangle essentially defines its shape; another piece of information is needed to give it scale. 1), but because dealing with something completely ‘sizeless’ is mathematically awkward, a simple work-around is usually employed: a length related to the geometry is designated as the unit of measure for the other lengths.
This means that we are not necessarily looking for the best global optimum, but rather for one that best balances absolute performance and robustness to small changes in shape (say, as a result of wear) or operating conditions. Such design searches become very tricky if the response of the geometry itself (to changes in design variables) can be abrupt – it will be difficult to tell whether this lack of robustness originates from the objective function or the geometry parameterization. 2), we may also consider the problem from the reverse standpoint; that is, the relationship between the location of any point of the geometry and the design variables should be differentiable.
For a semi-monocoque fuselage enclosing a pressurized cabin and/or a payload bay, any shape other than a cylinder with hemispherical end-caps is a stress engineering compromise. g. 5 , Z ∈ [0, 1]. 15) This is a somewhat unusual definition for a (semi-)circle, but it offers a simple way of increasing the flexibility of the geometry – at the expense of introducing additional design variables, of course. Here is what we might term a ‘generalized circle’:8 Y(Z) = Z N1 (1 − Z)N2 , 8 Z ∈ [0, 1]. 16) This is indeed a generalized circle – one may stumble upon other such geometrical artifices in unexpected places.
Aircraft Aerodynamic. Design Geometry and Optimization by Andras Sobester, Alexander I J Forrester