Optimization and Computational Fluid Dynamics - Department of ...

138 Nicolas R. Gauger
solution of the equations
� �
∂I ∂R ∂S
+ ψT + φT =0,
∂w ∂w ∂w
(5.41)
� �
∂I ∂R ∂S
+ ψT + φT =0.
∂Z ∂Z ∂Z
(5.42)
These are the adjoint equations for the problem of coupled aeroelasticity.
After their solution, the gradient can be recovered from the expression
� �
∂I ∂R ∂S
δI = + ψT + φT δX . (5.43)
∂X ∂X ∂X
with
We can assume the cost function to be a functional in the form
�
I(X,w,Z)= i(X,w,Z)dV (5.44)
i(X,w,Z)= Cp
Cref
V
(nx cosα + ny sinα)δ(η) (5.45)
where δ(η) is the Dirac delta function. The equation η = 0 defines the airfoil
shape in the body fitted coordinates (ξ,η). For the Dirac delta function under
integration the following equation holds
�
δ(η)f(η)dη =f(0) . (5.46)
In the context of Eq. (5.44), it reduces the volume integral to a surface
integral. We suppose that the fluid obeys the Euler equations, which in body
fitted coordinates take the form
∂F
∂ξ
+ ∂G
∂η
=0, (5.47)
where the transformed F,G are appropriate combinations of f and g, e.g.,
F = J ∂ξ
⎡
ρU
∂ξ ⎢
f + J g = J ⎢
ρuU +
∂x ∂y ⎣
∂ξ
∂xp ρvU + ∂ξ
∂yp ⎤
⎥
⎦ .
ρHU
(5.48)
Since our cost function I is of the form shown in Eq. (5.44), as first step we
have to formulate Eqs. (5.41) and (5.42) in an appropriate way, using the
following property