###### Abstract  
Numerical simulations are increasingly employed in the automotive industry to optimize the design stage, reduce prototype testing, and shorten the time to market. The aim of the presented research is the development of a fast and reliable method for the prediction of the sound field generated outside a vehicle by vibrating panels under electromagnetic shaker excitation. Despite that multi-physics numerical simulation software already link mechanical vibrations to their acoustic effect, they show a drawback when calculating the exterior sound field produced by a vibrating panel: the presence of a car model to separate front and rear radiations avoiding the acoustic short circuit, and an air volume surrounding it are required, thus increasing the model complexity and calculation time. Both problems can be overcome with the presented methodology: only the mechanical vibration of the panel is solved numerically, and the radiated sound field is then calculated postprocessing, relying on [PERSON]'s integral. At first, the method's validation is presented through laboratory experiments; then, a real vehicle panel is analyzed. Comparisons between the finite element method (FEM) simulations and experimental measurements showed very good agreement while keeping the calculation time low for both the laboratory and on-vehicle tests.  
automotive industry; clamped plates; controlled vibration; finite element method; electrodynamic shaker; laser doppler vibrometer; numerical simulation; vehicle panel +
Footnote †: journal: Applied Sciences  
Correlation (DIC) [18,19]. They can both be adopted to study vibrations, even though usually DIC is mainly used for structural testing and material identification [20]. At the same time, numerical simulations are more and more employed by car producers for improving the design of acoustics systems [21,22], reducing the need for expensive prototypes and avoiding time-consuming measurements. Therefore, simulation and processing methods capable of reducing the calculation time and providing high-accuracy results are strongly demanded. A common problem deals with the separation between the front and back radiation of a panel to avoid the acoustic short circuit effect [23]. In [24], the authors solved the problem by using an \"internal sound hard boundary\", accordingly to [25]. However, this solution can be applied only to simple geometries with the boundary lying on a plane similar to a flat plate. This work, instead, deals with a car bonnet, which would require to introduce in the numerical model the whole car, or at least its front part, to accomplish such requirements. In addition, an air volume surrounding the car should be included too as the simulation domain. Considering the typical dimensions of a car, this would result in a huge number of mesh elements, leading to an unacceptable computational time.  
In this paper, a fast and reliable mixed approach (numerical and analytical) to predict the exterior sound field radiated by a vibrating panel of a car under stationary forced excitation is presented. To overcome the aforementioned limitations, the proposed solution relies on an implicit FEM simulation only to solve the mechanical vibration of the panel. Then, the radiated sound field is obtained by calculating the Rayleigh's integral [26] during postprocessing. In a future development, the authors will consider explicit FEM simulations to analyze more advanced effects related to multiple reflection-refraction of the sound waves in the panels [27,28]. An extensive description of the proposed method is presented in Section 2. In Section 3, the results are shown subdivided between laboratory validation (Section 3.1) and on-vehicle tests (Section 3.2). Eventually, conclusions are summarized in Section 4.