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5 Efficient Deterministic Approaches for Aerodynamic Shape Optimization 113
in aerodynamic shape optimization by Jameson in his works on potential
equations [19] and later also for the Euler [21] as well as Navier-Stokes [20]
equations. Its main advantage over the finite differences is the significant increase
in speed since the corresponding numerical effort is independent of the
number of design variables. However, the implementation of the continuous
adjoint approach may be time-consuming and error-prone.
On the other hand, in the discrete case, one takes the discretized flow
equations for the derivation of the discrete adjoint problem (see e.g., [3, 12]).
This can be automated by so-called algorithmic, or automatic, differentiation
(AD) tools.
AD is a comparatively new field of mathematical sciences. This technique
is based on the observation that various elemental operations (like +, −, ×)
build up the cost function as their concatenation. Therefore, applying the
chain rule to this concatenation results in an automated differentiation of the
cost function. Depending on the starting point of the differentiation process –
either at the beginning or at the end of the respective chain of concatenations
– one distinguishes between the forward mode and the reverse mode of AD.
Using the reverse mode of AD, gradients can be computed very accurately at
a computational cost that is independent of the number of design variables.
This is the reason why this method is also called a discrete adjoint method.
At DLR, a few attempts have been made in AD computations. One attempt
is the differentiation of the DLR TAUij code by ADOL-C [14], which
is presented in this chapter as part of a differentiated optimization chain.
In the next sections, the different adjoint approaches will be explained
for single disciplinary aerodynamic shape optimization first and then their
extension to multidisciplinary design optimization (MDO) problems will be
discussed for aero-structure cases.
Finally, we will discuss the so-called one-shot methods. Here one breaks
open the simulation loop for optimization.
To start out, we explain how one can parameterize aerodynamic shapes
by the use of deformation techniques.
5.2 Parameterization by Deformation
In aerodynamic shape optimization, a geometry is either given by a parameterization
or can be changed by parameterized deformation. This means that
based on these parameters, a shape can be built up or deformed by a design
vector. Furthermore, the obtained shape has some aerodynamic properties
like the drag coefficient or pressure distribution. Therefore, the task of the
aerodynamic shape optimization is to optimize this design vector and its
dependent shape for some aerodynamic cost function.
When optimizing, there must be some chain to calculate the cost function
value at a given parameterization. This can be done by deforming a static