## 5 Application in a Case Study: Use of MPPs as a Ceiling Acoustic Treatment
The contour map shows the edge diffraction due to the seats, which are typically the only irregular element in a lecture hall contributing to the sound field diffusion. Figure 16 also shows a qualitative comparison with the standard GA simulation that involves a simpler modeling of the seats' blocks and a scattering coefficient (see Figure 16a). The DL-MPP corresponding to Configuration A in Table 4 was introduced in the model replacing most of the part of the existing false ceiling, as can be seen in Figure 15. Typically, the acoustic impedances needed as boundary conditions for an FDTD numerical simulation derive from the large amount of random incidence absorption coefficient datasets available through optimization processes [56]. Indeed, uncertainties inherent in energy parameters (absorption coefficients)--also due to ISO 354 limits [57]--are propagated in the conversion to non-unique corresponding complex acoustic impedances [58]. In this case, it has been  
Figure 16: Qualitative visualization of sound wave propagation within the lecture hall by means of GA (**a**) and FDTD (**b**) simulations throughout the longitudinal section. The sound source is located at the teacher’s position (S1 in Figure 14). Figure 15: Exterior (**left**) and interior (**right**) view of the acoustic treatment virtually introduced into the CAD model. The ceiling-mounted MPPs are highlighted in red. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Materials** & **125 Hz** & **250 Hz** & **500 Hz** & **1000 Hz** & **2000 Hz** & **4000 Hz** \\ \hline \hline \multirow{2}{  
possible to avoid some of those typical uncertainties in the workflow by directly starting with complex acoustic impedances values as input boundary conditions (see Figure 17) [59; 60]. Concerning the locally reactive absorption properties of the MPPs, the outcomes of the compensation model described in Section 3.3 were used to fit complex impedances to a boundary impedance model comprising a parallel network of second-order series RLC circuits [61]. The fitting procedure optimized non-negative circuit parameters (resistances, inductances and capacitances) in order to minimize the Euclidean norm of the error between the complex admittance of the circuit network and that of the data. A similar process was carried for known absorption coefficients of the remaining materials, following [62]. The complex admittance data and the model approximation to be used in the FDTD simulation are shown in Figure 17. As can be seen in the figure, the filter model described in [61; 62] and used in the FDTD simulation faithfully reproduced the complex admittance of the MPP material. The FDTD method chosen simulated up to 8000 Hz with 6.75 points per wavelength (Cartesian grid spacing \(h=6.35\,\mathrm{mm}\) and time step \(k=18.5\,\mathrm{\SIUnitSymbolMicro s}\)), which required approximately 2 h per sound source using four Nvidia Titan X (Maxwell) GPUs. It should be noticed that even though the FDTD model chosen in this work can switch to a ray-tracing algorithm over a certain frequency [63], in this case it was possible to run a full wave-based simulation thanks to the moderate complexity of the geometry and the availability of high computational power. Before introducing the double-layer MPP into the virtual model of the lecture hall, the calibration process was carried out based on the main room criteria acquired from the acoustic measurements. The 3D model was tuned in terms of reverberation time (T\({}_{30}\)) considering the sound source in location S1 and averaging the values over all the receiver points shown in Figure 14. Table 6 reports the trend in frequency of the measured values, the equivalent values derived from the calibrated FDTD model (\"without MPP\") and the variations due to the introduction of MPPs (\"with MPP\"). During the tuning of the 3D model, the tolerance range for the discrepancies between measured and simulated values was kept equal to twice the common JND [50], considering 10% of the measured values according to recent remarks [64].