Optimization and Computational Fluid Dynamics - Department of ...

88 Kyriakos C. Giannakoglou and Dimitrios I. Papadimitriou
δU
=
δbi
∂U
+
∂bi
∂U δxm
. (4.15)
∂xm δbi
This alternative formulation of the direct approach skips the computation of
grid sensitivities at the interior, since Eq. (4.12) depends only on δp
at the δbi
boundaries. However, as stated in the previous section, the CPU cost of the
direct approach is high since N solutions of the linearized flow equations are
necessary.
To set up the continuous adjoint approach, the augmented objective function
Faug is introduced by adding to F the field integral of the adjoint variables
multiplied by the flow equations. One may directly express the gradient
of F with respect to bi, by using either the total or the partial sensitivity
derivatives of the flow equations. The two alternative expressions are
δFaug
=
δbi
δF
� � �
T δ ∂fk
+ Ψ dΩ (4.16a)
δbi Ω δbi ∂xk
δFaug
=
δbi
δF
� � �
T ∂ ∂fk
+ Ψ dΩ . (4.16b)
δbi Ω ∂bi ∂xk
Starting from Eq. (4.16a), using the transformation from δ
� �
∂U to
δbi ∂xk
�
, [32], based on equation
∂
∂xk
� δU
δbi
δ
δbi
� �
∂U
∂xk
= ∂
∂xk
� δU
δbi
�
+ ∂U
∂ξ m
δ
δbi
� � m ∂ξ
∂xk
where ξ j are structured grid metrics and by integrating by parts, δF
δbi
(4.17)
is expressed
as follows
δF
=
δbi
1
�
�
2 δ(dS) T ∂fk
(p − ptar) − Ψ
2 Sw δbi Ω ∂ξm � � m δ ∂ξ
dΩ
δbi ∂xk
�
+ (Ψk+1p − Ψ T fk) δ(nkdS)
. (4.18)
δbi
Sw
Adjoint variables are computed by solving the field adjoint equations
∂Ψ
∂t − Ak T ∂Ψ
∂xk
with appropriate boundary conditions along the solid walls
and the inlet/outlet boundary
= 0 (4.19)
(p − ptar)+Ψk+1nk = 0 (4.20)
δU T
(A
δbi
T nΨ) = 0 (4.21)