Optimization and Computational Fluid Dynamics - Department of ...

96 Kyriakos C. Giannakoglou and Dimitrios I. Papadimitriou
Equations (4.45) can be solved to provide d2 U
dbidbj
at the cost of N(N+1)
2
equiv-
alent flow solutions. Instead, one may form the second derivative of the (new)
augmented objective function ˆ F by introducing (new) adjoint variables ˆ Ψn,
as follows
d 2 ˆ F
dbidbj
= d2 F
dbidbj
d 2 Rn
+ ˆ Ψn . (4.46)
dbidbj
Substituting Eqs. (4.44) and (4.45) in Eq. (4.46) and rearranging the terms
we obtain
d 2 ˆ F
dbidbj
∂ 2 Rn
= ∂2F +
∂bi∂bj
ˆ Ψn
∂bi∂bj
+ ∂2F dUk
+
∂bi∂Uk dbj
ˆ Ψn
�
∂F
+ +
∂Uk
ˆ ∂Rn
Ψn
∂Uk
+ ∂2F dUk
∂Uk∂Um dbi
∂ 2 Rn
∂bi∂Uk
� d 2 Uk
dbidbj
dUk
dbj
+ ∂2 F
∂Uk∂bj
∂ 2 Rn
dUm
+
dbj
ˆ dUk dUm
Ψn
∂Uk∂Um dbi dbj
∂ 2 Rn
dUk
+
dbi
ˆ dUk
Ψn
∂Uk∂bj dbi
. (4.47)
The term depending on the second derivatives of the flow variables (last term)
is eliminated by satisfying the adjoint equations
∂F
∂Uk
+ ˆ ∂Rn
Ψn = 0 (4.48)
∂Uk
at the cost of only one equivalent flow solution. The Hessian matrix is finally
expressed as follows
d 2 ˆ F
dbidbj
= ∂2F +
∂bi∂bj
ˆ ∂
Ψn
2 � 2
Rn ∂ F
+ +
∂bi∂bj ∂Uk∂Um
ˆ ∂
Ψn
2 �
Rn dUk dUm
∂Uk∂Um dbi dbj
� 2 ∂ F
+ +
∂bi∂Uk
ˆ ∂
Ψn
2 �
Rn dUk
∂bi∂Uk dbj
� 2 ∂ F
+ +
∂Uk∂bj
ˆ ∂
Ψn
2 �
Rn dUk
. (4.49)
∂Uk∂bj dbi
Thus, using the so-called direct-adjoint approach, the total CPU cost for a
Newton optimization cycle, Eqs. (4.43), is equal to 1+N+1 equivalent flow
solutions (including the solution of the flow equations).
4.6.2 Continuous Direct-adjoint Approach for the
Hessian (Inverse Design)
In the continuous approach, the gradient of F is given by Eq. (4.12). Starting
from Eq. (4.12), the second derivative of F is expressed as follows