  The modern aerodynamic optimization has a strong demand for parametric
methods with high levels of intuitiveness, flexibility, and representative
accuracy, which cannot be fully achieved through traditional airfoil parametric
techniques. In this paper, two deep learning-based generative schemes are
proposed to effectively capture the complexity of the design space while
satisfying specific constraints. 1. Soft-constrained scheme: a Conditional
Variational Autoencoder (CVAE)-based model to train geometric constraints as
part of the network directly. 2. Hard-constrained scheme: a VAE-based model to
generate diverse airfoils and an FFD-based technique to project the generated
airfoils onto the given constraints. According to the statistical results, the
reconstructed airfoils are both accurate and smooth, without any need for
additional filters. The soft-constrained scheme generates airfoils that exhibit
slight deviations from the expected geometric constraints, yet still converge
to the reference airfoil in both geometry space and objective space with some
degree of distribution bias. In contrast, the hard-constrained scheme produces
airfoils with a wider range of geometric diversity while strictly adhering to
the geometric constraints. The corresponding distribution in the objective
space is also more diverse, with isotropic uniformity around the reference
point and no significant bias. These proposed airfoil parametric methods can
break through the boundaries of training data in the objective space, providing
higher quality samples for random sampling and improving the efficiency of
optimization design.
