Optimization and Computational Fluid Dynamics - Department of ...

94 Kyriakos C. Giannakoglou and Dimitrios I. Papadimitriou
�
�
δF � � T δ(nkdS)
=− Ψ fk − Ψ
δbi Sw δbi Sw
T
� inv ∂f k −
∂xl
∂fvis
�
k δxl
nkdS
∂xl δbi
�
� � �
δ(nmdS) ∂uk ∂Ψm+1
+ ΨΛqm − μ +
Sw δbi Sw∂xl
∂xk
∂Ψk+1
� �
∂Ψq+1 δxl
+λδkm nmdS
∂xm ∂xq δbi
�
�
∂um δxl 1
−2 τkm nk dS+
∂xl δbi T τkm
∂uk δxl
nldS (4.40)
∂xm δbi
Sw
Sw
where Ψ satisfies the field adjoint equations
∂Ψ
∂t − AT k
∂Ψ
∂xk
The vector of source terms L is defined as
L1=− 1
T 2τkm
∂uk p
∂xm ρ2 (γ − 1)
− M −T K − M −T L = 0 . (4.41)
,Lk+1=2 ∂
∂xm
�
μ
T τkm
�
,LΛ=− ρ
p L1 .(4.42)
The boundary conditions over the solid wall are the same to those used for
the total pressure losses minimization; at the inlet and outlet boundaries,
Eq. (4.21) is imposed.
From Eq. (4.40), it can be stated that the gradient of F does not depend
on field integrals, even if F is, in fact, a field integral retaining thus, the
advantages of better accuracy and lower computational cost.
4.6 Computation of the Hessian Matrix
Starting from either the direct or the adjoint approach to compute first derivatives,
the use of the direct or adjoint approach anew leads to the computation
of the Hessian matrix. Consequently, four approaches to compute the exact
Hessian matrices have been developed. These approaches (either discrete or
continuous) differ with respect to their CPU cost.
Once the N(N+1)
2 distinct values of the symmetrical Hessian matrix as
well as the gradient of the objective function have been computed, the design
variables are updated using the (exact) Newton algorithm
d2F λ
db λ i
dbidbj
= −dF λ
dbj
(4.43a)
b λ+1
i = bλ i + dbλ i , i=1, ..., N (4.43b)
where λ is the optimization cycle counter.
Considering that the adjoint approach is much less time-consuming than
the direct approach for the gradient computation, one may think that the
exclusive use of adjoints would be advantageous for the computation of the