## 3 Numerical Simulations
### The Measurement Point Arrangement Determined by SVM  
When designing the measurement points on the surface of the structure, the measurement points are usually distributed on the structural surface uniformly in conventional strategies. In the theory of BEM, the interval between two adjacent measurement points should not be larger than one-sixth of the wavelength [16]. However, the criterion is not universal, especially when the frequency is quite low. Because the sound radiation at low frequencies is more of a concern in this paper, the theory of SVM should be adopted to be the guidance. In the work [7], [PERSON] pointed out that the vibration sampling interval is closely linked with the structural wavenumber, orientations of the observation points, and the target prediction error. Take a cylindrical shell with finite length as the example, the numbers of vibration sampling points along the axial direction and the circumferential direction obey different rules. It could be mainly summed up that, for most observation points in the sound field, no matter whether along the axial direction or the circumferential direction, the vibration sampling points should at least satisfy the Nyquist sampling law, and the number of measurement points should not be less than twice the modal order in this direction. Especially in the axial direction, the measurement points are usually denser. The number of axial measurement points is nearly directly proportional to the axial modal order, \(M_{a}\approx A_{0 err}n_{a}\). \(M_{a}\) is the axial measurement points and \(n_{a}\) is the axial modal order. \(A_{0 err}\) is a coefficient related to the level of intended prediction error and is given in Table 1.  
According to the study of [PERSON] [37], the value of \(A_{0 err}\) is usually chosen to be 3-4 in engineering to ensure the accuracy of sound prediction. Suppose that the structure is a cylindrical shell with fixed ends. The length of the cylindrical shell is 10 m, the radius is 1 m, and the thickness is 0.01 m. The material of the shell is steel, with a density of 7800 kg/m\({}^{3}\), [PERSON]'s modulus of \(2.06\times 10^{11}\) Pa, and Poisson's ratio of 0.3. The detail of structural modes could be calculated by COMSOL Multiphysics. When at different frequencies, the modal response could be extremely different. Even with the same excitation frequency, the existence of the reflection boundary could change the added mass of water to the structure. Thus, the natural frequency of the structure for every structural mode may change, which means that the modal response could change. Some modes are given out in Table 2. In the pictures, a dark color corresponds to a large displacement. In the study [17], [PERSON] pointed out that the low-order modes have an obvious characteristic of overall vibration in a large area, which leads to their higher sound radiation efficiency. However, in Table 2, it could be found that, at higher-order modes, the radiation efficiency could still not be neglected as the large displacement areas occupy most of the radiation surface. According to the theory of modal superposition, the displacement of the shell could be expressed in the form of modal superposition. In order to complete the sound prediction accurately, all the vibration modes should be collected with the measurement system. Therefore, the mode with the highest order determines the criterion of vibration sampling, which means that the intervals between two adjacent measurement points depend on the structural wavenumbers along the certain directions. \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Intended Error (dB)** & **0.1** & **0.2** & **0.4** & **0.8** & **1** & **2** & **3** \\ \hline \(A_{0 err}\) & 6.0 & 4.24 & 3.0 & 2.12 & 1.94 & 1.36 & 1.12 \\ \hline \end{tabular}
\end{table} 
Table 1: Values of \(A_{0 err}\) with respect to the intended prediction errors. In Table 2, \(h_{U}\) is the submerged depth of the structure. The modal natural frequency is \(f_{n}\) and the order of structural mode is \((n_{a},n_{c})\).