  In aerodynamic shape optimization, the convergence and computational cost are
greatly affected by the representation capacity and compactness of the design
space. Previous research has demonstrated that using a deep generative model to
parameterize two-dimensional (2D) airfoils achieves high representation
capacity/compactness, which significantly benefits shape optimization. In this
paper, we propose a deep generative model, Free-Form Deformation Generative
Adversarial Networks (FFD-GAN), that provides an efficient parameterization for
three-dimensional (3D) aerodynamic/hydrodynamic shapes like aircraft wings,
turbine blades, car bodies, and hulls. The learned model maps a compact set of
design variables to 3D surface points representing the shape. We ensure the
surface smoothness and continuity of generated geometries by incorporating an
FFD layer into the generative model. We demonstrate FFD-GAN's performance using
a wing shape design example. The results show that FFD-GAN can generate
realistic designs and form a reasonable parameterization. We further
demonstrate FFD-GAN's high representation compactness and capacity by testing
its design space coverage, the feasibility ratio of the design space, and its
performance in design optimization. We demonstrate that over 94% feasibility
ratio is achieved among wings randomly generated by the FFD-GAN, while FFD and
B-spline only achieve less than 31%. We also show that the FFD-GAN leads to an
order of magnitude faster convergence in a wing shape optimization problem,
compared to the FFD and the B-spline parameterizations.
