## 2 Materials and Methods
A B&K calibration system for accelerometers type 4294 was placed under the sensor head and measured for 20 s to calibrate the laser. It produces a pure tone at 159.2 Hz with a velocity of 10 mm/s (140 dB re 1 nm/s). For the shaker calibration, a pure tone at 159.2 Hz was played through it at 1 V, and a 20-s recording of the vibrating plate was taken with the laser. A MATLAB code was employed for generating the scanning grid and performing the whole measurement. A regular squared grid with a resolution of 30 mm was created, resulting in a total number of \(N\) = 77 points for a measurement time of 20\({}^{\prime}\) (5 s of stabilization delay were employed after every movement of the sensor head). The FEM model of the rectangular plate was solved in COMSOL Multiphysics, employing shell elements and a quad mesh with the same size of the scanning grid (Figure 4) for a total number of elements equal to 96 and \(N\) = 77 nodes. The solution was calculated with a frequency resolution of 1 Hz in the whole frequency range of interest, which was 150 Hz-1 kHz, for a total calculation time of 2\({}^{\prime}\)15\({}^{\prime\prime}\). The excitation was a unit force of 1 N applied vertically in the center of the plate (same position of the experimental measurements). The condition of a fully clamped plate was imposed on the edges. COMSOL built-in material properties were used for aluminum: \(E=70\) GPa, \(\
u=0.33\), \(\rho=2700\) kg/m\({}^{3}\). Damping was introduced as an isotropic loss factor of value \(\eta=0.015\), manually tuned against measurements. Figure 3: Block diagram of the experimental measurement setup. The numerical solution is evaluated in all the nodes of the mesh, thus allowing to obtain an output matrix having dimensions (\(N\times K\)) where \(N\) = 77 is the number of nodes and \(K\) = 851 is the number of simulated frequencies. The velocity data \(v\) was exported to have compatible inputs with the measured data. The velocity will be introduced in Equation (2) for solving the Rayleigh's integral. In Figure 5a, a surface plot of the velocity magnitude on the plate at 1 kHz can be seen, while, in Figure 5b, a grid of \(N\) = 77 points is shown where the solution is evaluated. It must be noted that directly calculating the SPL in COMSOL Multiphysics at the standard distance of 1 m would require a spherical air volume model around the plate, as in [24]. This would have led to a number of elements equal to 226,120 (instead of 96) due to the tetrahedral mesh of the air domain, and a calculation time of 1 h 20\({}^{\prime}\) (instead of 2\({}^{\prime}\)15\({}^{\prime}\)). The experimental measurements on a real vehicle panel were carried out at McLaren Automotive facilities. Two measurement sets were taken, with a mini-shaker mounted on the aluminum bonnet of a vehicle in two different positions. During the first measurement, the car hood was subdivided into a regular squared grid made of \(N\) = 368 nodes with 50-mm resolution (Figure 6a). The shaker was positioned in the bottom-left corner (Figure 7a), while four monoaxial piezoelectric accelerometers, having a sensitivity of 100 mV/g, were positioned in the nodes of the \"mesh\", four at a time (Figure 7b). This provided the acceleration data that will be employed to solve the [PERSON]'s integral with Equation (4). The system was driven by the same Zoom F8 soundcard and amplifier previously described for the laboratory experiment, and the same ESS of 10 s was employed. In addition, a B&K microphone was positioned at the standard distance of 1 m, on-axis with respect to the center of the bonnet. The aim of this first test was providing an experimental verification of the method in a real case, comparing the sound pressure with its estimation based on the Rayleigh integral and experimentally measured data. Figure 4: FEM model of the aluminum plate; the exciting point is highlighted in red. Figure 5: (**a**) Velocity magnitude over the plate at 1 kHz. (**b**) Grid of \(N\) = 77 points where the velocity data are evaluated.