Optimization and Computational Fluid Dynamics - Department of ...

8 Dominique Thévenin
Objective
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Point 2
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Fig. 1.4 Schematic representation of a simple optimization problem involving a single
parameter and a single objective. Now, the objective function shows two minima. The
points really known through a CFD-based evaluation are shown using symbols on top of
the (in fact unknown) objective function represented by the solid line in the background.
A gradient-based algorithm starting at Point 1 would probably find the right optimum
(symbols ∗) while the same algorithm starting from Point 2 (symbols +) would most
probably get stuck in the local minimum
discretization errors, etc. Such problems are often to be expected since
the numerical cost of the evaluations is the main problem for CFD-O as
explained at the end of the present Introduction. Therefore, in order to
facilitate CFD-O, many users will try to speed-up the evaluation process
as much as possible, thus using coarse grids or a very limited number of
iterations. As a consequence, the evaluations will be typically associated
with a certain amount of inaccuracy, which will depend on the configuration
considered. Obviously, it should remain small enough to allow for a
meaningful optimization. In many cases, it will lead to both a systematic
and to a stochastic evaluation uncertainty as depicted in Fig. 1.5.
The systematic error will move the optimal solution obtained by CFD-O
slightly away from the truly physical, optimal solution. There is no possibility
to solve this problem in principle, apart from using an extremely accurate
numerical procedure. The stochastic error will lead to a specific difficulty:
the appearance of a potentially high number of non-physical, local minima.
It becomes therefore even more importanttobeabletodealwiththisissue
in practice, in order to come near enough to the true optimal solution.