Efficient multi-stage aerodynamic topology optimization using an operator-based analytical differentiation

A high-performance density-based topology optimization tool is presented for laminar flows with focus on 2D and 3D aerodynamic problems via OpenFOAM software. Density-based methods are generally robust in terms of initial design, making them suitable for designing purposes. However, these methods require relatively fine resolutions for external flow problems to accurately capture the solid-fluid interfaces on Cartesian meshes, which makes them computationally very expensive, particularly for 3D problems. To address such high computational costs, two techniques are developed here. Firstly, an operator-based analytical differentiation (OAD) is proposed, which efficiently computes the exact partial derivatives of the flow solver (simpleFOAM). OAD also facilitates a convenient development process by minimizing hand-coding and utilizing the chain-rule technique, in contrast to full hand-differentiation, which is very complex and prone to implementation errors. Secondly, a multi-stage design process is proposed to further reduce the computational costs. In this technique, instead of using a fixed refined mesh, the optimization processes are initiated with a coarse mesh, and the converged solutions are projected to a locally refined mesh (as an initial guess) for a secondary optimization stage, which can be repeated to obtain a sufficient accuracy. A set of 2D and 3D laminar aerodynamic problems were studied, which promisingly confirmed the utility of the present approach, which can be adopted as a starting point for developing a design tool for large-scale aerodynamic engineering applications. In addition, the 3D problems indicated that less than 3 % of total optimization CPU-time is devoted to OAD, and multi-staging up to 45 % has reduced the overall costs.