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node functions and to their architecture, several ANN can be constructed. In the present work, a neural network based on radial basis functions is adopted [80]. The network is formed by an input layer, and intermediate layer, and an output layer. Values of the design variables are introduced in the input layer, whereas a single value, i.e. the fitness of the individual, is obtained as a linear combination of the intermediate layer values and returned as the network output. The weights of the linear combination are determined through a training procedure, and the number of neurons involved is taken equal to the number m of individuals of the training set. The approximation function associated to the jth neuron of the intermediate layer is of the form: ⎡ ⎛E ⎞2⎤ hj(x)=exp⎢−⎜ j ⎟ ⎥ (Eq.14) Where x is the input design variable vector, E j = x − x j , is the distance of the individual represented by x from the jth individual of the training set, and r is an attenuation coefficient. The fitness of x is then computed through the linear relation: m ⎢⎝r⎠⎥ ⎣⎦ f(x)=∑ω h (x) j j (Eq.15) j=1 The weights ωj are determined by requiring that the approximation function f satisfies the interpolation conditions: yk =f(xk) ∀k=1,...,m (Eq.16) where the {yk } represent the (known) values of the fitness fuctions associated to individuals of the training set. With this requirement, the training problem reduces to the solution of a linear system of m equations in m unknowns ωj. 3.3.2. Richardson Extrapolation In order to evaluate the difficulty for a Genetic Algorithm to converge, the fitness distance correlation method of [81] is considered. Let us consider a fitness function f: f:s∈S⊆Rn →f(s)∈R (Eq.17) (with S the search space and s an ordered N-ple of input variables defining a given individual) characterized, without loss of generality, by a unique global optimum s*. The fitness distance correlation (FDC) of f with respect to the fitness information in a discrete sample of the search space P ⊂ S is defined as: 44PDF Image | ANALYSIS AND OPTIMIZATION OF DENSE GAS FLOWS: APPLICATION TO ORGANIC RANKINE CYCLES TURBINES
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