## 3 Results
### Optimized MPP-SBHs  
It has been demonstrated in Section 3.3 that the increase in the rib density in SBHs results in increased absorbing performance due to additional acoustic losses. If an MPP is incorporated into the structure of the SBH, it provides additional losses, and the structure of the resulting MPP-SBH does not need to be very fine to provide reasonable absorbing performance. In all the following numerical examples, the MPP's thickness is fixed as \(t=0.2\,\mathrm{mm}\), and its holes' diameters \(d\) and the porosity \(\phi\) are the subject of the optimization together with the shape function. Figure 10 shows the optimization results in the case of na MPP-SBH with \(n=1\,\mathrm{cm}^{-1}\) (\(N=25\)), \(h_{\mathrm{r}}~{}=~{}1\,\mathrm{mm}\), and \(A_{\mathrm{min}}~{}=~{}0.25\,\mathrm{mm}\). Panel (a) of the figure depicts the optimal shape function together with the linear and quadratic one for the sake of a comparison. Similarly as in the case of the optimized SBHs, the optimal shape function for an MPP-SBH possesses small slopes at its extremities, protracting the most contracted part in the vicinity of the apex and non-unity value at the opening, namely \(G(1)=0.675\), introducing a cross-section jump. The diameters of the holes in the optimized MPP are \(0.17\,\mathrm{mm}\), with perforation ratio \(\phi~{}=~{}0.72\,\%\). Panel (b) of Figure 10 shows thee absorption coefficient spectra of the individual MPP-SBHs (in the case of the linear and quadratic profile, only the MPP parameters were optimized to maximize the value of \(\langle\mathcal{A}\rangle\)). The solid lines correspond to the numerical results of the 1D Riccati model, the dashed lines correspond to the 2D LNSE-FEM simulation for the validation; it can be observed that both the models essentially provide the same results. Similarly as in the case of the optimized SBH (see Figure 5), the MPP-SBH with an optimized  
Figure 10: Panel (a): shape functions. Panel (b): absorption coefficient spectra of MPP-SBHs with linear, quadratic, and optimized profiles. Solid lines—prediction by the Riccati model. Dashed lines—results from the LNSE-FEM model. In all the cases, \(A_{\mathrm{min}}=0.25\,\mathrm{mm}\), \(n~{}=~{}1\,\mathrm{cm}^{-1}\), and \(h_{\mathrm{r}}~{}=~{}1\,\mathrm{mm}\). Figure 9: Panel (a): shape functions of SBHs optimized for different lengths \(L\); in all the cases, \(A_{\mathrm{min}}~{}=~{}0.25\,\mathrm{mm}\), \(n~{}=~{}20\,\mathrm{cm}^{-1}\), and \(\xi=0.5\). Panel (b): the absorption coefficient spectra of the individual SBHs. shape provides the highest value of the mean absorption coefficient, namely \(\langle\mathcal{A}\rangle~{}=~{}0.883\). Both the absorption coefficient spectra in the case of quadratic and optimized MPP-SBHs possess characteristic low-frequency absorption peaks connected with the global resonance along the structures. In the case of the optimized MPP-SBHs, this peak has the value \(\mathcal{A}~{}=~{}0.900\) at a frequency of \(150\,\mathrm{Hz}\). All the shapes provide uniform and high absorption at higher frequencies. The distribution of the acoustic pressure amplitude in the optimized MPP-SBH discussed above at four different frequencies is shown in Figure 11. In all the cases, acoustic wave with amplitude \(1\,\mathrm{Pa}\) impinges on the MPP-SBH from the right. At a frequency of \(150\,\mathrm{Hz}\), corresponding to the low-frequency absorption peak (see Figure 10b), the maximum amplitude is at the MPP-SBH's apex, and the acoustic pressure amplitude quickly decreases with increasing longitudinal coordinate \(x\); meanwhile, the acoustic pressure does not depend on the transversal coordinate \(y\). This corroborates the low-frequency absorption peak results from the acoustic black hole effect. At a higher frequency of \(270\,\mathrm{Hz}\), the acoustic field becomes rather complex; there is non-zero acoustic pressure amplitude at the structure's apex as well as in the middle part, where it depends on the transversal coordinate \(y\) in slits.