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

TL;DR

This paper clarifies the concept of adjoint vectors in CFD, focusing on their use in output-based mesh adaptation and addressing the differences between continuous and discrete adjoints.

Contribution

It provides a clear explanation of adjoint concepts in CFD, unifying various notations and definitions to aid understanding and application.

Findings

Clarifies the distinction between continuous and discrete adjoints

Demonstrates the use of adjoints in mesh adaptation for accuracy

Provides a reference for adjoint application in CFD

Abstract

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.