## 2 Materials and Methods
In the second measurement, the shaker was positioned in the center of the bonnet (Figure 6b). The panel was excited with the same equipment, but only the B&K microphone was recorded, positioned at the standard distance of 1 m, on-axis with respect to the center. The panel was simulated numerically with the shaker in the same center position for obtaining the acceleration in a grid of points and then reconstructing the sound pressure at the virtual microphone position using the Rayleigh integral (Equation (4)). The aim of this second test was to compare the measured acoustic response of the panel with the numerically simulated result. The FEM model of the bonnet was solved in COMSOL Multiphysics (Figure 8) in the frequency range 150 Hz-1 kHz with a frequency resolution of 2 Hz. The three-dimensional geometry was simplified to two layers, modeled with shell elements having same thickness of the real panels, which was 0.9 mm for the inferior layer and 1.1 mm for the upper one. A triangular mesh was employed, with a minimum size defined as:  
\[mesh\_size_{min}=c_{0}/(6\cdot f_{max}) \tag{1}\]  
where \(c_{0}=343\) m/s is the speed of sound in air at 20 \({}^{\circ}\)C, \(f_{max}=1\) kHz is the maximum interested frequency and factor 6 represents the number of elements per wavelength, where a minimum number of 5 was suggested in [25]. The total number of elements of the mesh was 35,250, for a total simulation time of 1 h 30\({}^{\prime}\). The excitation was a unit force of 1 N applied vertically in correspondence to the red dot visible in Figure 8a. It corresponded to the same position of the shaker on the panel in the second experimental measurements (Figure 6b). The constraints were applied to the edges of the bottom layer (blue dots in  
Figure 6: (**a**) Test vehicle bonnet subdivided into a grid of \(N\) = 368 measurement points. (**b**) Scheme of second measurement shaker’s position (low center) of the bonnet. Figure 7: (**a**) Details of the test vehicle bonnet, with a mini-shaker on the bottom-left. (**b**) Details of the test vehicle bonnet with four accelerometers on the top-right. Figure 8b) in the corresponding positions of the hinges of the real vehicle. The material properties provided by the car manufacturer were: \(E=69\) Gpa, \(\
u=0.33\), \(\rho=2710\) kg/m\({}^{3}\). Additionally, in this case, an isotropic loss factor \(\eta=0.015\) was used for damping. As previously described for the plate, the numerical solution was evaluated in the same nodes of the measurement grid, thus allowing to obtain an output matrix with dimensions (\(N\times K\)), where \(N=368\) is the number of nodes and \(K=426\) is the number of simulated frequencies. The acceleration data \(a\) were exported, being compatible with the measured data. The acceleration was employed in Equation (4) for solving [PERSON]'s integral. In Figure 9a, a surface plot of the velocity magnitude on the bonnet at 1 kHz can be seen, while, in Figure 9b, the grid of \(N=368\) points is shown where the solution is evaluated. It this case, a direct calculation of the SPL in COMSOL Multiphysics at the standard distance of 1 m would have led to a number of elements of about 830,000 (instead of 35,250), considering a spherical air volume around the bonnet, for an estimated calculation time of about 32 h (instead of 1 h 30\({}^{\prime}\)). Moreover, this evaluation is strongly underestimated, since the whole car cockpit (or at least the front part of it) should be included in the model, according to the previous explanation, further increasing the number of elements and, thus, the calculation time.