Optimization and Computational Fluid Dynamics - Department of ...

14 Dominique Thévenin
Let us come now to the second possibility mentioned previously: reduce
as much as possible the needed number of evaluations. Here again, at least
two different sub-methods can be identified for this purpose:
• the first one consists in replacing a CFD-based evaluation by an appropriate
approximation of it. Depending on how this is done, this could also
be implemented as a model reduction and is thus related to the first item
listed in the previous list. But using Artificial Neural Networks, as done
in the Chapter 6 of this book, or some other kind of intelligent “interpolation”
in the broadest sense like the Response Surface Technique could
be seen indeed as a reduction of the number of needed CFD-evaluations.
As demonstrated in the present book, this method can be very efficient
to speed-up the full optimization procedure. Nevertheless, it is obvious
that great care must be taken, since the alternative evaluation method
should not falsify in any way the evolution of the optimization algorithm.
It is therefore not at all a trivial task to identify suitable approximation
methods!
• the second one leads again back to applied mathematicians. They have already
recognized a long time ago the issues associated with a large number
of evaluations, and indeed proposed an alternative formulation, “optimal”
from the point of view of numerical analysis: the adjoint method. Dueto
its importance for many practical problems, in particular in the aerospace
industry, three chapters of this book (3 to 5) will be mostly dedicated to
CFD-O based on adjoint methods. In principle, the adjoint approach requires
not much more than a “single evaluation”, which sounds almost too
good to be true! One difficulty of the adjoint approach is its formal complexity
for the common engineer, perhaps not familiar with all the needed
mathematical concepts. It is our hope that the corresponding chapters will
help such users understand how this method works. But two other major
difficulties are more or less inherent to the adjoint approach itself:
– a suitable consistent adjoint system must first be identified for the
considered system of equations. While this is easily done (and welldocumented
in the literature), e.g., for the Euler equations, the task
will become much more difficult when considering complex, multiphysics
problems involving perhaps a turbulent multiphase flow with chemical
reactions and concurrent objectives.
– furthermore, the adjoint approach requires a full knowledge of the intermediate
approximations on the way to the full solution of the system of
equations solved by CFD. In simple words, this means that the adjoint
approach, while reducing tremendously the requested number of evaluations
(and thus to a large extent the needed computing time), will lead
to a huge increase of the requested computer memory which will again
become a major problem for complex, three-dimensional flows involving
many unknowns at each discretization point.