• SQP
• Optimization
• PrandtlPlane
• smoothing technique
• non conventional configurations
• Mads
• direct search
• induced drag
• aerodynamics

The present work takes origin from studies carried out at the Department of
Aerospace Engineering of University of Pisa on innovative aircraft configurations.
In 1924 the German scientist and father of modern aerodynamics Prof. Ludwing
Prandtl demonstrated the existence of a wing system producing the minimum
induced drag, and he called it Best Wing System (BWS). This wing system is made
of four wings connected together at their tips so that it looks like a box in a frontal
view. The condition of minimum induced drag found by Prandtl states that the two
horizontal wings must have the same lift distribution whereas the vertical wings, or
bulkheads, must carry a butterfly shaped lift distribution. The induced drag of this
system is the lowest among all possible wings carrying the same total lift with a
given wingspan, and the efficiency increases as the vertical gap to wingspan ratio
( h b ) does. The Prandtl’s calculations were based on an approximated method
predicting an induced drag of the BWS as the non-dimensional gap h b! " that
is 0.16 times the reference monoplane induced drag. This is in contradiction with
the fact that when the two equally loaded horizontal wings are at infinite distance,
they can be treated as isolated, therefore they have half of the induced drag of the
equivalent monoplane.
In 1999 Prof. Aldo Frediani at University of Pisa gave a new demonstration based
on more accurate calculations. He obtained the same results as Prandtl in the
range of low values of h b , but now the asymptote is, according with theory, at 0.5.
At the same time, he started the translation into an engineering product of the
mathematical intuition of Prandtl and he called this aircraft PrandtlPlane in his
honor.
The PrandtlPlane is, hence, a new aircraft concept, based on the general aim to
reduce induced drag. Besides, this configuration has showed potential benefits in
different aircraft categories: from Very Large Aircraft (VLA) to small ULM airplanes.
The unconventional nature of this aircraft gives rise to new design challenges
mainly due to a lack of experience and of statistical data; the space of solutions is,
therefore, unexplored. For this reason a solution satisfying different conflicting
requirements such as wings with the same lift, static stability of flight and trim
condition in every point of the flight envelope may be hard to find.
The problem of finding the PrandtlPlane wing planform can be formulated in terms
of optimization, which is to find the geometric parameters defining the wings such
that the drag is minimized and the constraints are satisfied. Constraints involve the
equilibrium and the static stability during cruise and landing, maximum speed stall,
geometrical constraints, and so on. Boundaries on the variables are defined as
well: minimum/maximum swept angles, minimum/maximum dihedral angles,
minimum/maximum elevator and flaps deflections, and so on.
An overview of the main methods and algorithms for the search of minima of
unconstrained and constrained optimization problems is presented. Moreover, an
algorithm based on a derivative free method (MADS) for the search of minima of
unconstrained problems and its extension to constrained problems are presented.
An algorithm for the search of global minima is presented and it is tested on
benchmarking problems.
Finally, these methods are extensively applied to find solutions of a ULM version of
the PrandtlPlane aircraft and a flying model coming from the optimizer is
manufactured and tested.