Generative models offer design diversity but tend to be computationally expensive, while non-generative models are computationally cost-effective but produce less diverse and often invalid designs. However, the limitations of non-generative models can be overcome with the introduction of augmented shape signature vectors (SSVs) to represent both geometric and physical information. This recent advancement has inspired a systematic comparison of the effectiveness and efficiency of generative and non-generative models in constructing design spaces for novel and efficient design exploration and shape optimization, which is demonstrated in this work. These models are showcased in airfoil/hydrofoil design, and a comparison of the resulting design spaces is conducted in this work. A conventional generative adversarial network (GAN) and a state-of-the-art generative model, the performance-augmented diverse generative adversarial network (PaDGAN), are juxtaposed with a linear non-generative model based on the coupling of the Karhunen–Loève Expansion and a physics-informed shape signature vector (SSV-KLE). The comparison demonstrates that, with an appropriate shape encoding and a physics-augmented design space, non-generative models have the potential to cost-effectively generate high-performing valid designs with enhanced coverage of the design space. In this work, both approaches were applied to two large foil profile datasets comprising real-world and artificial designs generated through either a profile-generating parametric model or a deep-learning approach. These datasets were further enriched with integral properties of their members’ shapes, as well as physics-informed parameters. The obtained results illustrate that the design spaces constructed by the non-generative model outperform the generative model in terms of design validity, generating robust latent spaces with no or significantly fewer invalid designs when compared to generative models. The performance and diversity of the generated designs were compared to provide further insights about the quality of the resulting spaces. These findings can aid the engineering design community in making informed decisions when constructing design spaces for shape optimization, as it has been demonstrated that, under certain conditions, computationally inexpensive approaches can closely match or even outperform state-of-the art generative models.

Parametric models have been widely used in pertinent literature for reconstructing, modifying and representing a wide range of airfoil and/or hydrofoil profile geometries. Design spaces corresponding to these models can be exploited for modeling and profile-shape optimization under various performance criteria. Accuracy requirements, along with the need for modeling local features, often lead to high-dimensional design spaces that hinder the process of shape optimization and design through analysis. In this work, we propose a shape-informed dimensional reduction approach that attempts to tackle this deficiency by producing low-dimensional latent design spaces that can be efficiently used in shape representation and optimization. Furthermore, geometric moments are introduced in an attempt to cost-effectively capture analysis-relevant information that is generally expensive to produce. Specifically, geometric integral properties, although intrinsic features of the shape, are quite commonly related to performance indicators employed in performance optimization and therefore provide a cost-effective physics-informed component in the description of the design in the latent space. To this end, we employ the generalized Karhunen-Loève expansion to produce a shape- and physics-informed subspace retaining the highest possible geometric variance and robustness, that is, a lack of invalid designs. At the same time, a series of shape discretizations, encoding the foil’s shape profile, are examined with regard to their effect on the resulting latent space’s quality and efficiency. Our results demonstrate a significant reduction in the dimensionality of the original design space while maintaining a high representational capacity and a large percentage of valid geometries that facilitate fast convergence to optimal solutions in performance-based optimization.

This work investigates the use of physics-informed geometric operators in tandem with latent variable models to support surrogate learning, dimensionality reduction, and generative design of airfoils. A baseline dataset was constructed via a NURBS-based airfoil parametric model with physically interpretable design variables, and discretized using two schemes: a uniform parametric and a uniform arc-length sampling. A hybrid Variational AutoEncoder (VAE) with the addition of convolutional layers was developed and iteratively refined to evade shape invalidities. A systematic comparison of reconstruction accuracy, robustness, and diversity showed that a loss function based on the mean sum of squared distances had the best performance with sufficient stability during model optimization. However, this can be only established when the reconstruction and Kullback–Leibler terms in the $\beta$-VAE objective function are weighted via an appropriately selected $\beta$ value. Additionally, augmenting geometry with physics-correlated high-level descriptors, such as geometric moments, further improves latent-space quality. Among the tested operators, third-order geometric moments yielded the most consistent robustness gains. Discretization and achieved diversity proved to be linked, with uniform arc-length spacing achieving the best reconstruction accuracy, but with many near-identical designs that degraded the resulting diversity. In contrast, uniform parametric spacing exhibited higher diversity without the need for any special treatment of design distributions and diversity quantification measures. This study consolidates practical guidelines on architecture, loss-function scaling, physics-informed features, and quantification protocols for reliable, data-efficient airfoil generative design with VAEs.

Parametric models have been widely used in pertinent literature for reconstructing, modifying and representing a wide range of airfoil and/or hydrofoil profile geometries. Design spaces corresponding to these models can be exploited for modeling and profile-shape optimization under various performance criteria. Accuracy requirements, along with the need for modeling local features, often lead to high-dimensional design spaces that hinder the process of shape optimization and design through analysis. In this work, we propose a shape-informed dimensional reduction approach that attempts to tackle this deficiency by producing low-dimensional latent design spaces that can be efficiently used in shape representation and optimization. Furthermore, geometric moments are introduced in an attempt to cost-effectively capture analysis-relevant information that is generally expensive to produce. Specifically, geometric integral properties, although intrinsic features of the shape, are quite commonly related to performance indicators employed in performance optimization and therefore provide a cost-effective physics-informed component in the description of the design in the latent space. To this end, we employ the generalized Karhunen-Loeve expansion to produce a shape- and physics-informed subspace retaining the highest possible geometric variance and robustness, that is, lack of invalid designs. At the same time, a series of shape discretizations, encoding the foil's shape profile, are examined with regards to their effect on the resulting latent space's quality and efficiency. Our results demonstrate a significant reduction in the dimensionality of the original design space while maintaining a high representational capacity and a large percentage of valid geometries that facilitate fast convergence to optimal solutions in performance-based optimization.

In this work, we perform a systematic comparison of the effectiveness and efficiency of generative and non-generative models in constructing design spaces for novel and efficient design exploration and shape optimization. We apply these models in the case of airfoil/hydrofoil design and conduct the comparison on the resulting design spaces. A conventional Generative Adversarial Network (GAN) and a state-of-the-art generative model, the Performance-Augmented Diverse Generative Adversarial Network (PaDGAN), are juxtaposed with a linear non-generative model based on the coupling of the Karhunen-Loève Expansion and a physics-informed Shape Signature Vector (SSV-KLE). The comparison demonstrates that, with an appropriate shape encoding and a physics-augmented design space, non-generative models have the potential to cost-effectively generate high-performing valid designs with enhanced coverage of the design space. In this work, both approaches are applied to two large foil profile datasets comprising real-world and artificial designs generated through either a profile-generating parametric model or deep-learning approach. These datasets are further enriched with integral properties of their members' shapes as well as physics-informed parameters. Our results illustrate that the design spaces constructed by the non-generative model outperform the generative model in terms of design validity, generating robust latent spaces with none or significantly fewer invalid designs when compared to generative models. We aspire that these findings will aid the engineering design community in making informed decisions when constructing designs spaces for shape optimization, as we have show that under certain conditions computationally inexpensive approaches can closely match or even outperform state-of-the art generative models.