Optimization and Computational Fluid Dynamics - Department of ...

5 Efficient Deterministic Approaches for Aerodynamic Shape Optimization 135
Fig. 5.12 Wing structure model
Fig. 5.13 Validation of the aero-structural coupled adjoint with finite differences (AMP
wing, M∞ =0.78 and α =2.83 ◦ )
being (5.36) the flow and (5.37) the structural equations. We take the first
variation of the PDEs. This yields
δR = ∂R
∂X
δS = ∂S
∂X
δX + ∂R
∂w
δX + ∂S
∂w
∂R
δw + δZ =0, (5.38)
∂Z
∂S
δw + δZ =0. (5.39)
∂Z
We multiply Eqs. (5.38) and (5.39) with the Lagrange multipliers ψ and φ
respectively and add the result to the expression for the differential increment
of I in terms of the differentials of the independent set (X,w,Z), obtaining