# Vibroacoustic Optimization Study for the Volute Casing of a Centrifugal Fan
## 5 Vibroacoustic Optimization Strategy
### RBF Approximation Surrogate Models and Validation  
The high computational cost of each vibration and acoustic simulation to predict the vibroacoustics performance of each volume panel thickness design makes a direct optimization approach to find an optimum low-vibration noise solute unfeasible. Indeed, an optimized cycling takes 24-26 h. In addition, the optimization process terminates due to the failure of a single simulation program. Therefore, the long time of the optimal iterative procedure is the major bottleneck in the optimization of acoustics performance for all complex structures. Instead, the use of a surrogate model, which was constructed from and used in lieu of the actual simulation model, represents a valuable alternative to increase the speed of the optimization. For this reason, a metamodeling approach was chosen for vibroacoustic optimizations. Therefore, the combination of a design of experiment (DoE) and an approximation model (AM) was used to determine the relationship between the structural design variables and the target function, and provide the basic data and models for the subsequent optimization. According to the studied centrifugal fan volute, taking each volute thickness (FT, BT, and ST) as design variables, each design variable was given five levels. Table 3 provides the levels of distribution of the design variables. Due to the smaller number of design variables, the full-factor  
Figure 19: The volume parameterization structure. method was adopted to collect sample points of the design space constructed by the three variables collected; thus, a total of 125 sample points was collected. For each sample point, the aforementioned vibroacoustic coupling assessment method was used to calculate the radiated sound power and the total volute mass of the volute structure surface of each combination. The approximate model approach is generally divided into two broad categories: first, least squares fitting, also known as response surface methodology (RSM); and the second, interpolation. RSM uses polynomial functions to fit the design space. In addition, the RSM fitted the complex response relationships through regression models due to the simple algebraic expression. In addition, because of the advantages of being systematic and practical, the RSM had been used in a broad range [40; 41; 42; 43], but the approaching effect is not as good as that of neural network and other methods for highly complex functional relationships (a complex relation of nonlinear function). The radial basis function (RBF) belongs to the interpolation algorithms, and is the second approximate model method. Since the neural network model has a strong ability to approximate complex nonlinear functions, the learning speed is fast, has excellent generalization ability, and is highly fault-tolerant. It is used by many scholars and engineers in aircraft wingtips [44], highway traffics [45], turbomachinery [46], engineering optimization [47], compressors [48], gas cyclone separator [49], MMES controllers [50], and many other applications. The relationships between the vibrational sound radiation of the volute casing surface and the volute panel thickness are typically nonlinear. Thus, this study used the RBF model to approximate and fit the design space. Figure 20 shows the fitting procedure of the vibroacoustic optimization of volute casing. It can be seen from Figure 20 that this procedure is mainly divided into two parts. The first is the creation of a design space for the collected sample points using the assessment method of vibroacoustic coupling. The FEM model of fan volute was established by UG; then, ANSA constructed FEM mesh, and the Nasiran code solved the volute modal participation factor. At last, the vibrational sound radiation of volute was used by LMS Virtual Acoustics. In addition, all the calculation codes were integrated into the multi-disciplinary optimization platform, Isight. The second built an RBF approximation model (RBF surrogate model) instead of a simulation loop, as mentioned for the first part.