标题： 计算流体力学中伴随方程和输出误差估计简介 显示英文标题

标题： An Introduction to Adjoints and Output Error Estimation in Computational Fluid Dynamics

摘要： 近年来，伴随向量在计算流体力学（CFD）中的使用显著增加。 它们在许多应用中的作用，包括设计优化、数据同化和网格自适应，引起了研究人员和实践者的兴趣。 在这些领域中，伴随向量的概念被不同地解释，采用了各种符号和动机。 进一步复杂的是存在两种看似不同的伴随类型——“连续”和“离散”，以及在线性代数和泛函分析中采用的更正式的伴随算子定义。 这些问题可能使伴随的基本概念难以确定。 在这些笔记中，我们希望澄清一些与伴随向量相关的概念，并为连续和离散伴随提供一个有用的参考。 特别是，我们关注在基于输出的网格自适应中的伴随使用，其目标是通过有针对性的网格自适应来实现特定数量（或“输出”）的准确性。 虽然这是我们的研究应用，但这里讨论的概念直接适用于设计优化、数据同化以及许多其他使用伴随的领域。 显示英文摘要

摘要： In recent years, the use of adjoint vectors in Computational Fluid Dynamics (CFD) has seen a dramatic rise. Their utility in numerous applications, including design optimization, data assimilation, and mesh adaptation has sparked the interest of both researchers and practitioners alike. In many of these fields, the concept of an adjoint is explained differently, with various notations and motivations employed. Further complicating matters is the existence of two seemingly different types of adjoints -- "continuous" and "discrete" -- as well as the more formal definition of adjoint operators employed in linear algebra and functional analysis. These issues can make the fundamental concept of an adjoint difficult to pin down. In these notes, we hope to clarify some of the ideas surrounding adjoint vectors and to provide a useful reference for both continuous and discrete adjoints alike. In particular, we focus on the use of adjoints within the context of output-based mesh adaptation, where the goal is to achieve accuracy in a particular quantity (or "output") of interest by performing targeted adaptation of the computational mesh. While this is our application of interest, the ideas discussed here apply directly to design optimization, data assimilation, and many other fields where adjoints are employed.