Adjoint of a median-dual finite-volume scheme: Application to transonic aerodynamic shape optimization

Abstract

The sensitivity analysis is a crucial step in algorithms for gradient-based aerody- namic shape optimization. The analysis involves computing the gradient of function- als such as drag, lift, or aerodynamic moments, with respect to the parameters of the design. Gradients are efficiently calculated by solving adjoints of the linearized flow equations. The flow is modeled by the Euler equations of gas dynamics, solved in Edge, a Computational Fluid Dynamics (CFD) code for unstructured meshes. The adjoint equations and expressions for the gradients are derived here in the fully discrete case, that is, the mappings from the design variables to the functional’s values involve the discretized flow equations, a mesh deformation equation, and the parameterization of the geometry. We present a formalism and basic properties that enable a compact derivation of the adjoint for discretized flow equations obey- ing an edge-based structure, such as the vertex-centered median-dual finite volume discretization implemented in Edge. This approach is applied here to the optimiza- tion of the RAE 2822 airfoil and the ONERA M6 wing. In particular, we show a method to parameterize the shape, in 2D, in order to enforce smoothness and linear geometrical constraints.