On the one hand Sobolev gradient smoothing can considerably improve the performance of aerodynamic shape optimization and prevent issues with regularity. On the other hand Sobolev smoothing can also be interpreted as an approximation for the shape Hessian. This paper demonstrates, how Sobolev smoothing, interpreted as a shape Hessian approximation, offers considerable benefits, although the parameterization is smooth in itself already. Such an approach is especially beneficially in the context of simultaneous analysis and design, where we deal with inexact flow and adjoint solutions, also called One Shot optimization. Furthermore, the incorporation of the parameterization allows for direct application to engineering test cases, where shapes are always described by a CAD model. The new methodology presented in this paper is used for reference test cases from aerodynamic shape optimization and performance improvements in comparison to a classical Quasi-Newton scheme are shown.

翻译：一方面,Sobolev梯度的平滑可以大大改善空气动力形状优化的性能,防止出现规律性的问题。另一方面,Sobolev平滑也可以被解释为赫西安形状的近似值。本文展示了Sobolev平滑(被解释为赫西安形状的近似值)如何带来相当大的好处,尽管参数化本身已经很顺利。在同时分析和设计的背景下,这种方法特别有益,我们处理不精确的流量和联合解决方案,也称为“一个射击优化”。此外,纳入参数还允许直接应用工程测试案例,在这些案例中,形状总是由CAD模型描述。本文提出的新方法用于参考测试案例,从空气动力形状优化和性能改进到古典的 Quasi-Newton 方案。