Optimization and Computational Fluid Dynamics - Department of ...

4 Adjoint Methods for Shape Optimization 95
Hessian as well. However, if first-order sensitivity derivatives are computed
using the adjoint approach, N pairs of additional systems of direct or adjoint
equations must be solved for the computation of the Hessian matrix. So, the
total cost to compute all quantities needed by Eq. (4.43) is equal to that of
1+2N equivalent flow solutions.
The high cost of the previous approaches is due to the fact that the adjoint
approach is used for the first derivative and therefore, dU values are not
dbi
available; as it will become clear below, these quantities are necessary in
order to proceed to the computation of the Hessian matrix. For this reason,
the direct approach to compute the gradient of F must be employed. Given
this decision, one should investigate the expected gain from the subsequent
use of the adjoint approach to compute the Hessian (direct-adjoint approach)
compared to the direct-direct approach, where the computation of the Hessian
is based on the repeated application of the direct approach. It can be shown
(this development is omitted) that the direct-direct approach requires N +
N(N+1)
2 equivalent flow solutions at each optimization cycle. On the other
hand, the direct-adjoint approach, in which second derivatives are computed
using the adjoint approach, is the less time-consuming one since it requires
the solution of 1+N equivalent flow problems. Note that the previous figures
must be augmented by one, so as to account for the cost of solving the flow
equations and computing the value of F.
4.6.1 Discrete Direct-adjoint Approach for the Hessian
Starting from Eq. (4.1), the second derivative of F is given by
d 2 F
dbidbj
= ∂2F +
∂bi∂bj
∂2F ∂bi∂Uk
+ ∂2F dUk dUm
∂Uk∂Um dbi dbj
dUk
dbj
Equation (4.44) can be used to compute d2 F
dbidbj
ties) and d2 U
dbidbj
( N(N+1)
2
+ ∂2F dUk
∂Uk∂bj dbi
+ ∂F
∂Uk
d 2 Uk
dbidbj
provided that dU
dbi
. (4.44)
(N quanti-
quantities) can also be computed. The first derivatives
of U are computed at the cost of N equivalent flow solutions, Eq. (4.2),
according to the so-called direct approach. For the additional N(N+1)
2 quantities
(second derivatives of U), from Eq. (4.2), we get
d 2 Rn
dbidbj
= ∂2Rn +
∂bi∂bj
∂2Rn ∂bi∂Uk
+ ∂2Rn dUk
∂Uk∂Um dbi
dUm
dbj
dUk
dbj
+ ∂2Rn dUk
∂Uk∂bj dbi
+ ∂Rn
∂Uk
d 2 Uk
=0 . (4.45)
dbidbj