The paper "An Introduction to the Adjoint Approach to Design" by Michael B. Giles and Niles A. Pierce provides an overview of the adjoint method in computational fluid dynamics (CFD), particularly for aeronautical applications. The authors emphasize the simplicity of the adjoint approach when viewed through the lens of linear algebra. They discuss the extension of the adjoint method to partial differential equations (PDEs), the construction of the adjoint PDE and its boundary conditions, and the physical significance of the adjoint solution. The paper also covers examples of using adjoint methods to optimize the design of business jets. The introduction highlights the historical use of adjoint equations in optimal control theory and their application in fluid dynamics, starting with Pironneau's work and Jameson's pioneering contributions. Jameson developed the adjoint approach for various flow problems, progressing from 2D airfoil optimization to 3D wing design and complete aircraft configurations. The paper also mentions other research groups that have developed adjoint CFD codes for design optimization, including Elliott and Anderson's work on unstructured grids and Mohammadi's use of automatic differentiation. The authors address the limitations and complexity of the adjoint approach, particularly in the mathematical formulation and code creation. They aim to simplify these aspects by presenting the adjoint theory in the context of linear algebra, which forms the basis for the discrete adjoint CFD approach. The paper then discusses the extension to PDEs, focusing on the construction of the adjoint PDE and its boundary conditions, and the introduction of geometric perturbations. Finally, the paper concludes with a discussion of the pros and cons of the discrete and continuous approaches and provides examples of using adjoint methods to optimize business jet designs.The paper "An Introduction to the Adjoint Approach to Design" by Michael B. Giles and Niles A. Pierce provides an overview of the adjoint method in computational fluid dynamics (CFD), particularly for aeronautical applications. The authors emphasize the simplicity of the adjoint approach when viewed through the lens of linear algebra. They discuss the extension of the adjoint method to partial differential equations (PDEs), the construction of the adjoint PDE and its boundary conditions, and the physical significance of the adjoint solution. The paper also covers examples of using adjoint methods to optimize the design of business jets. The introduction highlights the historical use of adjoint equations in optimal control theory and their application in fluid dynamics, starting with Pironneau's work and Jameson's pioneering contributions. Jameson developed the adjoint approach for various flow problems, progressing from 2D airfoil optimization to 3D wing design and complete aircraft configurations. The paper also mentions other research groups that have developed adjoint CFD codes for design optimization, including Elliott and Anderson's work on unstructured grids and Mohammadi's use of automatic differentiation. The authors address the limitations and complexity of the adjoint approach, particularly in the mathematical formulation and code creation. They aim to simplify these aspects by presenting the adjoint theory in the context of linear algebra, which forms the basis for the discrete adjoint CFD approach. The paper then discusses the extension to PDEs, focusing on the construction of the adjoint PDE and its boundary conditions, and the introduction of geometric perturbations. Finally, the paper concludes with a discussion of the pros and cons of the discrete and continuous approaches and provides examples of using adjoint methods to optimize business jet designs.