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2 A Few Illustrative Examples of CFD-based Optimization 23
merical simulations based on RANS are still widely used today for engineering
problems and complex geometries due to a higher computational efficiency.
The objective in this case is to optimize the prediction of the time-averaged
turbulent velocity distribution in channel flows.
A similar investigation was performed in [37], where a variable Schmidt
number model for jet-in-crossflows was improved using GA optimization.
Multi-objective EAs have been also employed for determining and adjusting
reaction parameters in [18, 19, 20]. But, to our knowledge, no paper has
been published up to now considering the optimization of the model constants
of an engineering turbulence model.
2.2 Evolutionary Algorithms for Multi-objective
Optimization
2.2.1 Multi-objective Optimization
Mathematically speaking, a multi-objective problem consists of optimizing
(i.e., minimizing or maximizing) several objectives simultaneously, with a
number of inequality or equality constraints. The problem can be formally
written as follows:
subject to:
Find x =(xi) ∀ i =1, 2,...,Nparam such as
fi(x) is a minimum (resp. maximum), ∀ i =1, 2,...,Nobj
gj(x)=0, ∀ j =1, 2,...,M (2.1)
hk(x) ≤ 0, ∀ k =1, 2,...,K (2.2)
where x is a vector containing the Nparam design parameters, (fi)i=1...Nobj
the objective functions and Nobj the number of objectives. In this study, only
inequality constraints are considered and are prescribed as bounded domains.
In other words, upper and lower limits are imposed on all parameters:
xi ∈ [xi,min; xi,max] i =1...Nparam . (2.3)
The objective function (fi(x)) returns a vector containing the set
i=1...Nobj
of Nobj values associated with the elementary objectives to be optimized
simultaneously. The number of input parameters and objective functions for
the different cases considered in this chapter are summarized in Table 2.1.
Acommonpracticetosolvesuchaproblem is to use a trade-off between
the objectives by linearly combining them using some fixed weights prescribed
by the user (see for example [16, 73]). The resulting single objective function