## 2 Preamble
### Application for Squeal Noise and Convergence Results
#### 3.4.2 Application and Numerical Results  
Figures 7 and 8 give the evolution of \(\epsilon_{j}^{ac}\) at \(\mu=0.8\), depending of the size of the C&B basis as well as DMS basis for the acoustic patterns during the steady state vibration regime on the three horizontal planes (at 0.25  
2/--CB\" for j = 1 to 6). Of course, this conclusion is also clearly illustrated by showing the evolution of \(\lambda_{ac}\), as illustrated in Figure 9. It can also be observed that the convergence results are very similar for the two cases (i.e., acoustic patterns on the horizontal and vertical planes). These results indicate that increasing \(\gamma\) (i.e., the size of the C&B reduction) allows us to enhance the visualization (i.e., image) of the acoustic radiation and therefore the prediction of the squeal noise. In order to illustrate this fact, Figure 10 gives the acoustic patterns on one chosen horizontal plane (at the distance of 0.75 m in the direction z, the second horizontal plane in Figure 3) for the reference model (denoted \"Ref.\", first line of Figure 10) and two chosen C&B reductions (with \(\gamma=1\) and \(\gamma=1.5\) in the second and third lines of Figure 10, respectively). The result obtained for \(\gamma=1.5\) is very similar to the reference both in form and in acoustic level, while the results for \(\gamma=1\) are worse with an overestimation of the acoustic intensity. Then, in order to analyze only the effects of the DMS size, the reference is chosen as the numerical result of the C&B reduction technique with \(\gamma=1.5\) (i.e., this choice avoids introducing cumulative errors from the two successive C&B and DMS reductions in the value of \(\epsilon_{j}^{ac}\)). It should be noted that the reduced system \(\gamma=1.5\) comprises 2885 degrees of freedom, divided into 2640 degrees of freedom at the interfaces (i.e., spatial physical degrees of freedom), plus 245 generalized degrees of freedom corresponding to the modes with fixed interfaces (103 generalized degrees of freedom (dof) for the disc, 111 generalized dof for the upper pad and 31 generalized dof for the lower pad). Analyzing the convergence of the DMS basis in Figures 7 and 8 (see more specifically bars for the legends \"Scale 2/--DMS\" for j = 1 to 6), it is observed that increasing the number of interface modes (i.e., \(N_{DMS}\)) does not significantly decrease \(\epsilon_{j}^{ac}\). In fact this result demonstrates that considering few complex eigenmodes (i.e., \(N_{DMS}>50\)) seems to be enough to provide a satisfactory approximation of the radiated fields at different resolutions. Results for \(\lambda_{ac}\) reinforce this conclusion with a value of \(\lambda_{ac}\) inferior of \(8\times 10^{-2}\), whatever the size of the tested DMS basis. Note that, in the present case, convergence results for \(\epsilon_{j}^{ac}\) and \(\lambda_{ac}\) are provided by applying, beforehand, the C&B reduction technique with \(\gamma=1.5\). Figure 10 also gives the visualization of the acoustic radiation on the horizontal plane at the distance of 0.75 m for three DMS reductions (\(N_{DMS}=50\), \(N_{DMS}=150\) and \(N_{DMS}=1000\), the last three lines of Figure 10), to be compared with \(\gamma=1.5\) (i.e., the third line of Figure 10). In order to further illustrate the relevance of the proposed methodology, numerical simulations are carried out to estimate the radiated acoustic field for \(\mu=1\). Figure 11 gives the evolution of \(\lambda_{ac}\) for different C&B and DMS bases. Moreover Figure 10 gives the associated acoustic patterns for the reference model, two C&B bases and three DMS reductions (see the second column of Figure 10). All the results are in agreement with the discussion made previously for \(\mu=0.8\): increasing \(\gamma\) (i.e., the size of C&B basis) increases the quality of the estimation of the approximated acoustic noise, whereas increasing the number of complex interface modes (i.e., \(N_{DMS}\)) does not significantly improve the image quality of the radiated field.