# A Novel Optimization Framework to Replicate the Vibro-Acoustics Response of an Aircraft Fuselage
## 4 Optimization Process
### MDO Optimization Procedure  
The MDO optimization procedure proceeded by iterating the following steps:  
* generation of the interpolated response surfaces, based on the DOE table data;
* identification of the best model using cross-validation;
* run of the global optimization method on the response surface, to find a candidate point that minimizes the metrics (Equation (1));
* check of the proximity of the candidate point to the existing samples and, if needed, apply a random shift;
* space-filling, to balance for the global optimization (based on surrogate models with known data) to fall into local minima without performing additional exploration;
* run the FEM analyses on the two new points;
* update the initial DOE data and iterate until a terminate criterion is satisfied.  
Thus, the starting point of the MDO strategy is an existing DOE database, not necessarily tailored for optimization. Then, the global optimization method Differential Evolution (DEVOL) [14], was executed to find a \"good\" candidate point [12] on the Response Surface Method (RSM).  
From the algorithm perspective, there are no differences between a run on real analyses or on surrogated models, and this represents a trade-off between accuracy and simulation time. Models reproduce all the outputs of the simulation, including the constraints, in order to preserve results consistency.  
The used metamodel, interpolated Radial Basis Functions (RBFs), use multi-quadratic, linear, cubic and thin-plate kernels; other numerical formulations, like approximated or neural-network-based, have been considered but not included.  
Every time models are created or updated, their predictive capabilities were evaluated by means of cross-validation, using the R2 press regression parameter (PRESS stands for Predicted Residual Error Sum of Squares). Every model is repetitively generated for datasets with one experiment removed and then evaluated on the missing experiment, to compare predicted and simulated results. The results of all these calculations are added into one regression value that, if close to 1, indicates that the model will perform well for points that were not simulated. Only the best model with the highest R2 press was used for optimization. The automated validation and selection of the model with the higher predictive capability is part of the functionalities provided by the PIDO platform; the best kernel varied through the campaign but on average, cubic was the most exploited.  
Initial value of the cross-validation (DOE only) was low, 0.3 or less in a range between -1 and +1, where +1 indicates that the model's predictions always match perfectly the reference (the model is iteratively re-built without a sample and the prediction is compared with the sample) whereas post-optimization values were between 0.65 and 0.7. This value is still not enough to provide results without proper validation; all the candidate configurations are investigated numerically and not just on the metamodel.  
RBFs have been preferred over approximated ones (such as the least square) for the higher accuracy without significant overhead in their construction (in the order of a few seconds with 300 experiments, 8 inputs, 30 outputs), whereas like Kriging and Light Weight Neural Network had similar accuracy with slightly superior construction time.  
Models assembly time was important as this operation was performed for each iteration.