## 3 Numerical Simulations
### The Results of VR-ESM  
Repeat the simulations in Section 3.1. The quantity of measurement points is fixed at 90. The optimal values of \(L1\) are 42 and 35, which are determined by the method in Section 2.3. Define \(\gamma^{\prime}\) as the condition number of, \(\gamma^{\prime}=\text{cond}\Big{(}\widehat{\mathbf{\Phi}}\Big{)}\). The variation of \(\gamma^{\prime}\) versus the ARM order is shown in Figure 12a. The reconstruction errors versus the ARM order are shown in Figure 12b. A total of 32 measurement points are selected as test points, whose arrangement is shown in Figure 12c. For observing the measurement point arrangement conveniently, the shell is spread into the planar form along the circumferential direction as Figure 12c. Figure 11: The sound prediction errors of CMESM on the prediction plane when 135 uniformly arranged measurement points are used at the frequency of 71 Hz. (**a**) Free field (\(L1=92\), upper: \(p_{E}\); lower: \(\varphi_{E}\) ) and (**b**) \(h_{L1}=0.2\) m (\(L1=95\), upper: \(p_{E}\); lower: \(\varphi_{E}\) ). Figure 12: (**a**) The variation of \(\gamma^{\prime}\) with the increase of ARM order, (**b**) the variation of reconstruction error versus the ARM order (\(f_{0}=71\) Hz, \(h_{L1}=0.2\) m ), and (**c**) the arrangement of test points. From Figure 12a, it could be seen that, although \(\gamma^{\prime}\) increases with the increase of ARM order, the value of \(\gamma^{\prime}\) is quite low. Thus, the reconstruction of vibration could be quite robust. As shown in Figure 12b, the tendencies of \(\varepsilon_{v}\) versus the ARM order are similar for test points and total points, which manifests the efficiency of the reconstruction method. It could be seen that, with the increase of ARM order, \(\varepsilon_{v}\) decreases at the first \(L1\) orders to reach the minimum, which means that the computations reach the convergence when \(L1\) orders are used. However, when the ARM order continues to increase, \(\varepsilon_{v}\) begins to increase because \(\gamma^{\prime}\) also increases, which means that the prediction errors caused by the noise could be reinforced. Thus, it is still necessary to select the optimal \(L1\) to keep the prediction accurate. The prediction results and the measurement point arrangements are shown in Figures 13 and 14. The truncation numbers of ARMs are selected to be 42 and 35 for the free field and half-space, respectively. It could be seen that the prediction results are enough accurate with VR-ESM though the measurement points are quite sparse, which verifies the efficiency of VR-ESM. When the quantity of measurement points is fixed, compared with the uniform measurement point arrangement in CMESM, more accurate prediction results could be obtained with the VR-ESM when the measurement point arrangement is optimized with the EFI, as the independence among the measurement points is reinforced by minimizing the condition number of the sensing matrix. To verify the necessity of the optimization process with EFI, the prediction results of VR-ESM with uniform measurement points (axial: 6; circumferential: 15) are shown in Figure 15. The optimal \(L1\) are determined by minimizing \(\varepsilon_{v}\). Figure 14: The sound prediction errors of VR-ESM on the prediction plane when \(h_{U}=0.2\) m at the frequency of 71 Hz, \(L1=42\). (**a**) Upper: \(p_{E}\); lower: \(\varphi_{E}\). (**b**) The arrangement of measurement points (90 points optimized with EFI). Figure 13: The sound prediction errors of VR-ESM on the prediction plane in the free field at the frequency of 71 Hz, \(L1=42\). (**a**) Upper: \(p_{E}\); lower: \(\varphi_{E}\). (**b**) The arrangement of measurement points (90 points optimized with EFI). It could be seen that, with uniform and sparse measurement points, the prediction results are not accurate. The phenomenon could be explained by the ill-condition problem of the sensing matrices.