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fdcP ( f ) = s∈P 11 ∑[(f(s)− fP)(d(s,s*)−dP)] (Eq. 18) with fP = 1 ∑ f (s) and dP = 1 ∑d(s,s*) the mean fitness and the mean distance of P s∈P P s∈P the sample individuals from the global optimum, respectively. The following distance definition is used: ∆i with si the i-th input variable of the N-ple and ∆i its range of variation. If a maximization (respectively, minimization) problem is considered, the FDC returns a value of -1 for a fitness function of the form f =C1 −C2d(s,s*) (resp., f =C1 +C2d(s,s*)), (Eq.20) with C1,C2 ∈R,C2 >0. FDC measures the deviation of a fitness function from the class of strictly concave (resp., strictly convex) functions of the form of ((Eq. 20)): if the FDC of a given fitness function is close to -1 (resp., +1), this one can be easily optimized by a GA, whereas if FDC≈0 (no correlation with function (Eq. 20)), the function is GA-hard. The use of a more accurate scheme and of finer meshes allows reducing the GA-Hardness of an optimization problem. Unfortunately, the use of very fine meshes considerably increases computational costs. In order to alleviate this problem, a simple and effective method based on Richardson extrapolation is proposed (see [57]). Let us note f2h and fh two estimates of the fitness function f, respectively computed on a “coarse” mesh 2h and on a finer mesh h. The approximation error in the two cases is: ε = f − f =Chp +T.O.S.; ε = f − f =C(2h)p +H.O.T. (Eq.21) h exact h 2h exact 2h where p is the scheme’s convergence order, and C its error constant. Neglecting Higher Order Terms and combining the two equations above the following estimate of the exact solution is obtained: 45 ⎛∑ 2 ⎞2 ⎛ 2 ⎞2 ⎜ (f(s)−f)⎟⎜ (d(s,s*)−d)⎟ P ⎝s∈P ⎠ ⎝s∈P ⎠ ∑ P d(s,s*)= ∑ i i=1 i (Eq.19) N (s−s*)PDF Image | ANALYSIS AND OPTIMIZATION OF DENSE GAS FLOWS: APPLICATION TO ORGANIC RANKINE CYCLES TURBINES
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