Technical Report 2006-013

Adjoint of a Median-Dual Finite-Volume Scheme Applied to 2D and 3D Transonic Aerodynamic Shape Optimization

O. Amoignon and M. Berggren

March 2006

The sensitivity analysis is a crucial step in algorithms for gradient-based aerodynamic shape optimization. The analysis involves computing the gradient of functionals 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 obeying an edge-based structure, such as the vertex-centered median-dual finite volume discretization implemented in Edge. This approach is applied here to the optimization 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.

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