# Surrogate-Based Optimization of a Centrifugal Pump with
## 5 Optimization Strategy
### Surrogate-Based Optimization  
In literature, the single- (SOGA [18]) or multi-objective (MOGA [19]) genetic algorithms are used for turbomachinery shape optimization, due to their easiness and robustness: objective functions derivatives are not requested and the probability to remain trapped in a local optimum is very low. To alleviate the computational effort requested by genetic algorithms due to the large number of evaluations, the use of a surrogate model to approximate and evaluate the objective functions during the optimization process is mandatory. Both Kriging (KRG) [20,21] and artificial neural network (ANN) are used for the pumps optimization, even if [PERSON] et al. [18] show that KRG performs better than ANN in this context. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Variable** & **Baseline** & **Optimum** \\ \hline \multirow{8}{*}{Impeller} & \(D_{2}\) & 70.10 mm & 68.00 mm \\  & \(D_{0}\) & 41.29 mm & 40.05 mm \\  & \(b_{2}\) & 8.10 mm & 7.90 mm \\  & \(D_{1,h}\) & 22.73 mm & 26.4 mm \\  & \(D_{1,s}\) & 41.29 mm & 42.76 mm \\  & \(\beta_{b1,h}\) & 29.50\({}^{\circ}\) & 28.40\({}^{\circ}\) \\  & \(\beta_{b2,h}\) & 40.00\({}^{\circ}\) & 37.60\({}^{\circ}\) \\  & \(\beta_{b2,h}\) & 40.00\({}^{\circ}\) & 42.40\({}^{\circ}\) \\  & \(\gamma_{h}\) & 85.00\({}^{\circ}\) & 91.05\({}^{\circ}\) \\  & \(\gamma_{s}\) & 85.00\({}^{\circ}\) & 89.86\({}^{\circ}\) \\  & \(\delta_{s}\) & 0.00\({}^{\circ}\) & \(-1.33^{\circ}\) \\  & \(n_{b}\) & 6 & 7 \\ \hline \multirow{2}{*}{Vaneless diffuser} & \(D_{3}\) & 72.1 mm & 74.48 mm \\  & \(b_{3}\) & 8.1 mm & 10.75 mm \\ \hline \multirow{2}{*}{Volute} & \(A_{4}\) & 907 mm\({}^{2}\) & 1043 mm\({}^{2}\) \\  & \(\alpha_{i}\) & 30.00\({}^{\circ}\) & 30.60\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: List of the geometrical parameters for the baseline and optimized geometries provided by the LP code and the SBO algorithm, respectively. The KRG model performs a global approximation involving interpolation and the objective function is defined as:  
\[\hat{f}(\underline{x})=\underline{g}(\underline{x})^{T}\underline{\beta}+ \epsilon(\underline{x}),\]  
i.e., \(\hat{f}\) is the sum of trend basis functions (here polynomials), which fits the DoE, and a stochastic function \(\epsilon(x)\) with mean zero, which fixes the function to interpolate the experiments. \(\underline{g}(\underline{x})\) is the vector of the basis functions, \(\underline{\beta}\) is the vector of the generalized least squares estimates of the basis function coefficients and \(\epsilon(x)\) is computed using a Gaussian correlation function. The whole optimization strategy is managed by the Dakota [22] software and is summarized by the following steps:  
1. Computation of a Design of Experiments (DoE) to create a training points database. The DoE is generated using the Latin Hypercube Sampling (LHS) method, which allows to randomly and uniformly distribute the designs over the whole design space. The DoE consists of \(10N\) training points, where \(N\) is the number of design variables, as proposed by [PERSON] et al. [23]. 2. Training points evaluation. The points are evaluated using the CFD approach described in Section 4. 3. Surrogate models generation based on Kriging for the approximation of pump efficiency and pressure head. 4. Search for the maximum of the efficiency by means of the constrained SOGA applied to the surrogate model. Crossover rate and mutation rate are set equal to 0.8 and 0.1, respectively. 5. Verification of the maximum through a CFD simulation. 6. If the convergence criterion is not met, add the maximum to the training points database and return to step 3. The iterative process described above improves continuously the surrogate accuracy and accelerates the optimization convergence.