The purpose of this paper is the presentation of a technique to be integrated in a numerical airfoil optimization scheme, for the exact satisfaction of a strict equality cross-sectional area constraint.
An airfoil optimization framework is presented, based on Area-Preserving Free-Form Deformation (AP FFD) technique. A parallel metamodel-assisted differential evolution (DE) algorithm is used as an optimizer. In each generation of the DE algorithm, before the evaluation of the fitness function, AP FFD is applied to each candidate solution, via coupling a classic B-Spline-based FFD with an area correction step. The area correction step is achieved by solving a sub problem, which consists of computing and applying the minimum possible offset to each one of the free-to-move control points of the FFD lattice, subject to the area preservation constraint.
The proposed methodology is able to obtain better values of the objective function, compared to both a classic penalty function approach and a generic framework for handling constraints, which suggests the separation of constraints and objectives (separation-sub-swarm), without any loss of the convergence capabilities of the DE algorithm, while it also guarantees an exact area preservation. Due to the linearity of the area constraint in each axis, the extraction of an inexpensive closed-form solution to the sub problem is possible by using the method of Lagrange multipliers.
AP FFD can be easily incorporated into any 2D shape optimization/design process, as it is a time-saving and easy-to-implement repair algorithm, independent from the nature of the problem at hand.
The proposed methodology proved to be an efficient tool in facing airfoil design problems, enhancing the rigidity of the optimal airfoil by preserving its cross-sectional area to a predefined value.
