Discretization error control for constrained aerodynamic shape optimization

In this paper, gradient-based aerodynamic shape optimization with output constraints is implemented using adaptive meshes updated via adjoint-based error estimates. All the constraints, including geometry and trim conditions, are handled simultaneously in the optimization. The trim constraints may involve outputs that are not directly targeted for optimization, and hence also not for error estimation and mesh adaptation. However, numerical errors in these outputs often indirectly affect the calculation of the objective. The method adopted in this work takes this effect into account, so that the mesh is adapted to predict both objective outputs and constraint outputs with appropriate accuracy. In other words, the entire optimization problem is targeted for adaptation. Instead of optimizing with a single fixed mesh resolution, the objective function is first evaluated on a relatively coarse mesh, which is subsequently adapted as the shape optimization proceeds. As the shape approaches the optimal design, the mesh becomes finer, in necessary regions, leading to a multi-fidelity optimization process. The multi-fidelity framework saves computational resources by reducing the mesh size at early stages of optimization, when the design is far from optimal and most of the shape changes happen. The optimization at each fidelity terminates once the change of the objective is smaller than the tolerance (estimated error) at the current fidelity mesh, reducing unnecessary effort in optimizing on low-fidelity meshes. The use of output-based error estimates prevents over-optimizing on a coarse mesh, or over-refining on an undesired shape. The proposed framework is demonstrated on several optimization problems, starting with a NACA 0012 airfoil. In each case, the shape is optimized to minimize the drag given a target lift and a minimum airfoil area. The accuracy and efficiency of the proposed method are investigated through comparisons with existing optimization techniques. We expect the framework to be even more important for more complex configurations, dramatic shape changes, or high-accuracy requirements.

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