# A Novel Optimization Framework to Replicate the Vibro-Acoustics Response of an Aircraft Fuselage
## 4 Optimization Process
### Modeled Problem  
In the modeled problem, the sound power and phase of each monopole source represented the input data for the MDO process, whereas the Acoustic Intensities (AIs) at microphones provided the output. Thus, a parametrization of the model for the SCs was required to change the input values for the FEM simulations and to get the output to compare with RC data. Such parametrization of the model was implemented in the PIDO platform, using the related interfaces after an appropriate setup and validation of the FEM model.  
The interfaces are designed to simplify the access to the identified model parameters and, during the execution of the analyses, automatically map the desired values of the design variables to the model itself and extract the values of the outputs of interest. The information gathered through analyses are processed to determine the relationships between inputs and outputs; this knowledge can then be passed to different post-process tools or used to perform design optimization, reliability study, robust design.  
The same validated workflow with parametrized FEM model, analysis, results extraction and post-process has been used both to perform DOE and optimizations. DOEs have been run both to provide a clear overview of the design variables' impact and to generate a re-usable database of simulations. Due to the nature of the design variables (and specifically the periodicity of the phase) some of the DOE approaches would have been ineffective (in particular, the members of the factorial-class, that explore the combinations of the lower and higher values of the design variables, but also other methods like the Plackett-Burman are affected by the same limitations). The final objective was to determine configuration of the sources able to provide the best approximation of the known RC pressure field; this has been organized as an optimization problem where a dedicated performance indicator (Equation (1)), able to describe the difference between the reference and the simulated values, had to be minimized. A single averaged, performance indicator was selected.  
The problem to be solved was far from trivial; exploratory DOE and subsequent correlation studies highlighted that, although the amplitude of the sources had a relevant and, within certain conditions, predictable impact on the pressure field, the phase was significantly less manageable. PIDO platforms typically rely on black-box optimizers, algorithms that neither have insight on the problem physics, nor make assumptions about its continuity or smoothness.  
The initial studies underlined the non-linear and non-monotone behavior of the outputs with respect to design variables; from the optimization point of view, this creates several local minima that hinder the convergence of the algorithm.  
Global optimizers are methodologies that explore the whole design space, therefore they are suitable for this kind of problem; however, this exploration requires a relevant number of experiments [14]. On the other hand, local optimizers explore limited regions of the domain and often rely on the analytical or numerical evaluation of the gradient to determine the local solution. They are usually attracted by minima close to the starting point, whose setup becomes critical, although gradient-based line search can be exploited to extend the investigated portion of the domain.