A parametric level set method with convolutional encoder-decoder network for shape optimization with fluid flow

In this article, we present a new data-driven shape optimization approach for implicit hydrofoil morphing via a polynomial perturbation of parametric level set representation. Without introducing any change in topology, the hydrofoil morphing is achieved by six shape design variables associated with the amplitude and shape of the perturbed displacements. The proposed approach has three to four times lower design variables than shape optimization via free-form deformation techniques and almost two orders lower design variables compared to topology optimization via traditional parametric level sets. Using the fixed Cartesian level set mesh, we also integrate deep convolutional encoder-decoder networks as a surrogate of high-fidelity Reynolds-averaged Navier-Stokes (RANS) simulations for learning the flow field around hydrofoil shapes. We show that an efficient shape representation via parametric level sets can enable online convolutional encoder-decoder application for the shape optimization of hydrofoils. The generalized flow field prediction of the convolutional encoder-decoder is demonstrated by a mean and minimum structural similarity index measure of 0.985 and 0.95, respectively, for predicted solutions compared to RANS predictions for out-of-training shapes. The convolutional encoder-decoder predictions are performed nearly five orders of magnitude faster compared to RANS. This enables a computationally tractable surrogate-based drag minimization of fifty different hydrofoils for two different design lift coefficients. Furthermore, the best local minimum obtained via the surrogate-based optimization lie in the neighbourhood of the RANS-predicted counterparts for both the design lift cases. The present findings show promise for the future shape optimization via parametric level sets with convolutional encoder-decoder over a broader spectrum of flow conditions and shapes.