## 2 Preamble
### Prediction of Sausal Noise Based on Acoustic Radiation  
One of the most important goals in the field of friction-induced vibration is to be able to predict not only the stability and transient nonlinear behavior of the system but also the acoustic noise during brake squeal. To meet this goal, one of the most popular approaches for modeling the acoustic problem is based on the Boundary Element Method (BEM). The BEM can be decomposed into two main steps: first the surface sound pressure calculation \(\mathbf{P}_{S}^{\omega}\) which depends on the surface normal velocity field \(\dot{\mathbf{X}}_{\text{n}}\), and secondly the estimation of the free space sound pressure \(\mathbf{P}^{\omega}\) by calculating the surface sound pressure. The associated equation corresponding to the second step is defined by:  
\[\epsilon\mathbf{P}^{\omega}=\mathbb{M}^{\omega}\mathbf{P}_{S}^{\omega}- \mathbb{L}^{\omega}\frac{\partial\mathbf{P}_{S}^{\omega}}{\partial\mathbf{n}} \tag{8}\]  
where \(\mathbf{P}_{S}^{\omega}\) corresponds to the sound pressure over the system skin (denoted by \(\partial\Omega\)) and \(\omega\) defines the pulsation. The scalar \(\epsilon\) is in [0; 1] depending on the location of the calculated pressure \(\mathbf{P}^{\omega}\): \(\epsilon=1\) in the radiation space and \(\epsilon=\frac{1}{2}\) on boundaries). \(\mathbf{n}\) denotes the outside normal of \(\partial\Omega\). \(\mathbf{P}^{\omega}\) is the sound pressure in the free space and \(\mathbb{L}^{\omega}\) and \(\mathbb{M}^{\omega}\) are the BEM matrices (for more details see [26; 27]). It is important to note that the term \(\frac{\partial\mathbf{P}_{S}^{\omega}}{\partial\mathbf{n}}\) is a known value since it is directly linked to the surface normal velocity \(\dot{\mathbf{X}}_{\text{n}}\), which is an output value of the structural calculation.  
The previous fields depend on the pulsation \(\omega\). So, the BEM has to be applied for each harmonic component of the nonlinear response of the brake system in order to estimate the acoustic radiation. For the present study, the multifrequency acoustic calculation method [5], that allows us to calculate the global sound pressure (\(\mathbf{P}\) in the free space and \(\mathbf{P}_{S}\) over the mesh) by superposition of the BEM results for each frequency, is applied to estimate the noise levels over the mesh and at every point in free space. One of the main advantage of this methodology is that only the predominant contributions are used for the global estimation of the acoustic radiation. The pressure level in dB can be calculated, such as \(L_{\text{p,bem}}=10\log_{10}(\mathbf{P}^{*}/P_{ref}^{2})\), where \(P_{ref}=2\times 10^{-5}\) Pa and \({}^{*}\) denote the conjugated elements. For the system under study, it should be mentioned that we propose the boundary element model that is composed of the upper or inner part of the brake system skin for the calculation of the acoustic radiation.  
Figure 2: Transient nonlinear behavior of the system for \(\mu=0.8\) (**a**) time responses and (**b**) spectrograms.  
us to have a notion of convergence with respect to different patterns of the acoustic field emitted during squeal noise.  
First, a reminder of the notion of the windowed Fourier Transform as well as wavelets and their extension to the two-dimensional case is proposed. Then the proposed approach based on the multiresolution signal decomposition with the wavelet representation and the decomposition algorithm proposed by [PERSON] [28; 29] are developed. The use of such an approach for a convergence study of squeal noise is discussed and some convergence criteria are proposed for the analysis of acoustic fields. Finally, the efficiency of the proposed strategy is undertaken based on numerical results.