Optimization and Computational Fluid Dynamics - Department of ...

5 Efficient Deterministic Approaches for Aerodynamic Shape Optimization 127
design vector (X) initial surface (xs)
ց ւ
defgeo initial surface (xs)
↓ (x) ւ
difgeo initial computational grid (ms)
↓ (dx) ւ
meshdefo
↓ (m)
TAUij
↓ (CD)
Fig. 5.4 Chain to compute the cost function value
∂CD
∂X
= ∂CD
∂m
∂m ∂x
· ·
∂dx ∂X
. (5.32)
The optimization strategy in the following computations is a steepest descent
method which was implemented as an optimizer into the optimization
framework Synaps Pointer Pro. This framework has the possibility to read
in user-defined gradients. Therefore, the gradients are calculated by separate
routines and are then submitted to the optimizer.
5.6.1 Test Case Definition
As test case for the validation and application of AD generated adjoint sensitivity
calculations an RAE 2822 airfoil is chosen with a Mach number of 0.73
and an angle of attack of 2 ◦ . The drag coefficient for this test case has been
optimized with both parameterizations, Hicks-Henne and cosine function parameterizations
(see Sect. 5.2.1). In both optimizations, 20 design parameters
have been used. The computational grid has 161 × 33 grid points.
5.6.2 Finite Differences
To compute the finite differences in order to have a validation framework
for the AD sensitivities, the first task was to tune the stepsize h for the
approximation
∂I
∂xi
(X) ≈ I(X + hei) − I(X)
h
(1 ≤ i ≤ n) (5.33)