Aerodynamic optimal design is crucial for enhancing performance of aircrafts, while calculating multi-target functionals through solving dual equations with arbitrary right-hand sides remains challenging. In this paper, a novel multi-target framework of DWR-based mesh refinement is proposed and analyzed. Theoretically, an extrapolation method is generalized to expand multi-variable functionals, which guarantees the dual equations of different objective functionals can be calculated separately. Numerically, an algorithm of calculating multi-target functionals is designed based on the multi-mesh approach, which can help to obtain different dual solutions simultaneously. One feature of our framework is the algorithm is easy to implement with the help of the hierarchical geometry tree structure and the calculation avoids the Galerkin orthogonality naturally. The framework takes a balance between different targets even when they are not the same orders of magnitude. While existing approach uses a linear combination of different components in multi-target functionals for adaptation, it introduces additional coefficients for adjusting. With each component calculated under a dual-consistent scheme, this multi-mesh framework addresses challenges such as the lift-drag ratio and other kinds of multi-target functionals, ensuring smooth convergence and precise calculations of dual solutions.

翻译：气动优化设计对于提升飞行器性能至关重要，然而通过求解具有任意右侧项的对偶方程来计算多目标泛函仍具挑战性。本文提出并分析了一种基于对偶加权残差（DWR）网格细化的新型多目标框架。理论上，本文推广了一种外推法以展开多变量泛函，从而保证不同目标泛函的对偶方程可独立求解。数值上，基于多网格方法设计了一种计算多目标泛函的算法，该算法可同时获取不同的对偶解。本框架的一个特点是：借助层次化几何树结构，算法易于实现，且计算过程天然避免了Galerkin正交性。该框架能在不同目标之间取得平衡，即使其量级不同。现有方法采用多目标泛函中不同分量的线性组合进行自适应，但需引入额外调节系数。通过在对偶一致格式下计算各分量，本多网格框架解决了升阻比及其他类型多目标泛函的计算难题，确保了对偶解的平滑收敛与精确计算。