Optimization and Computational Fluid Dynamics - Department of ...

5 Efficient Deterministic Approaches for Aerodynamic Shape Optimization 121
and
δxξ,...,δyη =0,δw=0 on B = B(X) (5.21)
are also solved just once.
The right hand side −d(I) of the wall boundary condition of the quasiunsteady
adjoint Euler equations is dependent on the cost function I. The
adjoint far-field boundary condition just states that the geometrical position
of the far-field is fixed and free stream conditions apply there.
Finally, the components of the gradient ∇XI =(δiI)i=1,...,n can now be
determined via an integration just over the adjoint solution and the metric
sensitivities δxξ,...,δyη and
�
δI = − p(−ψ2δyξ + ψ3δxξ) dl + K(I)
C
�
−
ψ
D
⊤ ξ (δyηf − δxηg)+ψ ⊤ η (−δyξf + δxξg) dA (5.22)
is obtained where K(I) is again a term dependent on the cost function I.
For the gradient of the drag, the following right hand side adjoint boundary
on C is used
d(CD)=
2
γM 2 ∞ p∞Cref
(nx cosα + ny sinα) (5.23)
and to get the corresponding gradient, K(I) is
K(CD)= 1
�
Cp(δnx cosα + δny sinα) dl (5.24)
for the gradient of the lift
and
d(CL)=
Cref
K(CL)= 1
C
2
γM 2 ∞ p∞Cref
Cref
are used, and for the gradient of the pitching moment
and
d(Cm)=
2
�
C
γM 2 ∞ p∞C 2 ref
(ny cosα − nx sinα) (5.25)
Cp(δny cosα − δnx sinα) dl (5.26)
(ny(x − xm) − nx(y − ym)) (5.27)
K(Cm)= 1
C2 �
Cpδ(ny(x − xm) − nx(y − ym)) dl (5.28)
ref C
are used. For more details see [7] or [8].