# 3D Acoustic Mapping in Automotive Wind Tunnel: Algorithm and Problem Analysis on Simulated Data
## 5 Results of Parametric Study
### Resolution of Point Grid  
In this paragraph, three different grid steps have been tested with spans of 100 mm, 50 mm, and 25 mm. Noise-free simulations are used and the masking effect is not considered. Hereafter, the trimmed calculation grid is used. In Figure 11, a comparison  
Figure 10: Source power ratio \(W_{s}\). Ideal source power ratio is 0. (**a**) Full grid. (**b**) Trimmed grid. Figure 9: Scores of full versus trimmed grid of potential sources. (**a**) Non-dimensional distance error \(D_{s}\); average value obtained from identified sources only. Transparent bars represent the ratio between grid step (50 mm) and \(\lambda\). (**b**) Power concentration \(W_{\infty}\), the vertical axis is zoomed on the range of 99 to 100%. among maps obtained at different frequencies with different grid resolutions is shown. It is possible to see that grid refinement enables observing a clearer image of different sources. The quantitative comparison reported in terms of source localization (\(D_{s}\)) and power concentration (\(W_{\mathrm{s}}\)) in Figure 12 shows evident improvement in the localization accuracy passing from coarsest to finest grid. However, the use of a fine grid brings a major advantage only at high frequency. In any case, it is interesting to observe that average error is in the order of magnitude of 1/10 of the wavelength both with 50 mm and 25 mm of grid step. The power concentration \(W_{\mathrm{s}}\) of the map is good for all cases shown here, except for 2500 Hz and 100 mm of grid step. In this case, most of the source energy is outside the region of integration, because the wavelength, i.e., the radius of the sphere to define the zone of interest, is only slightly greater than the grid step. Indeed, even if the sources are correctly detected, as depicted in Figure 11e, the equivalent sources are not concentrated enough to produce an high score. The choice of \(p=0\) tends to concentrate the equivalent source power in few potential sources near the actual one, but, depending on noise and other instabilities, the equivalent sources may be shifted of one grid step (or more). This, combined with the big grid step, leads to bad scores in this case. Regarding source quantification, the only evident negative effect happens in the case already mentioned. Therefore, it is possible to conclude that the grid step should be a small fraction of the wavelength of analysis to obtain accurate results. However, the assessment of this fraction is very difficult, because results obtained with inverse methods are strongly dependent on the particular problem definition and the assumption made. As rule of thumb, it is possible to claim that at least two point sources per wavelength must be used in the grid. The accuracy on the identification of the acoustic power emitted by each source is reported in Figure 13 in terms of source power ratio (\(W_{\mathrm{s}}\)). As expected, the coarse resolution of the calculation volume induces an important underestimation of the source power, especially in the high-frequency range (see the square markers in in Figure 13c representing \(W_{\mathrm{s}}\) for the frequency band of 2500 Hz). The major impact of the grid size is on computation time, as shown in Figure 14. As the maps are calculated with the same software and hardware, the computation time is calculated as the average of all spectral lines. Making a comparison when the grid size changes, it is possible to recognize a linear behavior with the number of points. An anomalous behavior, i.e., over the linear trend, is shown by the full grid: despite the better regularization condition, the full grid contains a lot of meaningless points (those inside the model), needing a larger computational effort. Figure 11: Comparison among maps obtained at different frequencies with the finest and the coarsest grid resolution. (**a**) 800 Hz and 100 mm. (**b**) 800 Hz and 25 mm. (**c**) 1600 Hz and 100 mm. (**d**) 1600 Hz and 25 mm. (**e**) 2500 Hz and 100 mm. (**f**) 2500 Hz and 25 mm.