## 3 Case Study
### Model Analysis  
The prerequisite for the OSP of arch dams is to conduct a natural frequency analysis, and the influence of reservoir water on natural frequencies cannot be ignored. Many scholars have employed the added mass method proposed by [PERSON] to simulate the effect of reservoir water on the upstream surface of a dam body. However, this method neglects the compressibility of the reservoir water, leading to an overestimation of the calculated natural frequencies. To simulate the dynamic characteristics of the dam more realistically, two methods, Westergaard added mass model and FSCM, are used to simulate the dynamic water pressure of the reservoir water, which is divided into four models for discussion, namely, an empty reservoir model I without considering the reservoir water, an Westergaard added mass model II with a reduction factor of 0.5, an incompressible fluid solid coupling model III, and a compressible fluid solid coupling model IV. Because the optimal sensor placement is based on the modal information calculated through dynamic characteristic analysis of the model, and the model established based on FSCM is more reasonable, the results of sensor optimization arrangement are also more reasonable. Calculated natural frequencies and mode shapes are presented in Table 3 and Figure 6. Table 3 presents the natural frequencies of the first 20 modal orders calculated, while Figure 6 shows the first 5 modal orders. Modal order refers to the number of eigenvalues obtained by a structure in modal analysis. In finite element analysis, the essence of modal analysis is to find the eigenvalues of a matrix, and the number of eigenvalues is the order of the modal. Due to the infinite dimensionality of the actual analysis object, its modals have infinite orders, but in calculations, only the first few modal orders are usually considered. This article focuses on the first 20 modal orders. Figure 5: Arch dam model and crack elements: (**a**) the on-site image of the arch dam, (**b**) arch-dam–reservoir–foundation system, (**c**) dam body (C3D8), (**d**) reservoir water (AC3D8), (**e**) schematic diagram of contraction joints, and (**f**) schematic diagram of all cracks (COH3D8). According to Table 3, natural frequencies of models II, III, and IV are lower compared to model I. This indicates that the presence of reservoir water causes a decrease in the natural frequencies of the arch dam. This is consistent with the principle that, when considering reservoir water, the system's mass increases while the stiffness remains constant, leading to a reduction in natural frequencies [39]. Model IV has the lowest natural frequency, suggesting that considering the compressibility of reservoir water has the most significant impact on natural frequencies of the arch dam. Based on Figure 6, it is evident that the modal shapes vary when considering the compressibility of reservoir water, compared to the other three models. Taking the fourth modal shape as an example, model IV exhibits a symmetric modal shape, while the other three models display anti-symmetric shapes. This indicates that considering the compressibility of reservoir water has a certain effect on the modal shapes of the arch dam. In summary, when conducting natural frequency analysis