## 3 Case Study
### OSP Considering the Evolution of Structural States and Material Properties  
Arch dams are subjected to long-term loads and environmental effects during operation, which lead to the evolution of structural states and material properties, including the degradation of contraction joints, the appearance of cracks, and elastic modulus zoning  
Figure 15: Selection of optimal number of sensors. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Algorithm** & **PSO** & **IPSO** & **SSA** & **ISSAMS** & **AHA** & **IAHA** \\ \hline Run time/s & 202.15 & 202.60 & 99.99 & 106.07 & 101.32 & 101.43 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Average running time of each algorithm. degradation of the dam body. The evolution of structural states and material properties in an arch dam can reduce the dam's stiffness, resulting in a decrease in the system's natural frequencies and subsequently affecting the placement of sensors. To investigate the impact of factors such as cracks, contraction joints, and elastic modulus zoning degradation of the dam body on the placement of sensors on the downstream surface of the arch dam, 18 different cases are set up for calculation and analysis. The specific settings for the 18 cases are provided in Table 6.  
The contact elastic modulus of contraction joints significantly influences the natural frequencies of arch dams [42]. To study the effect of different elastic moduli of contraction joints on OSP, the elastic moduli of the contraction joints are taken as k = 0.25 GPa, 0.5 GPa, 1 GPa, 2 GPa, 10 GPa, and 20 GPa. Figure 16 shows the first-order natural frequency curves for different elastic moduli of contraction joints. As the elastic modulus of the contraction joint increase, the natural frequency of the arch dam significantly increases. The most noticeable change in natural frequencies occurs in the range of about 0.25 to 2 GPa, becomes gradual in the range of about 2 to 10 GPa, and stabilizes in the range of about 10 to 20 GPa. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & **Cracks, Contraction Joints, and Elastic Modulus Degradation** & **Schematic Diagram of Cracks** & **Cracks, Contraction Joints, and Elastic Modulus Degradation** & **Schematic Diagram of Cracks** \\ \hline
1 & Horizontal cracks & 10 & Dam’s elastic modulus zoning degradation for 20 years \\
2 & Vertical cracks & 11 & Dam’s elastic modulus zoning degradation for 30 years \\
3 & Horizontal and Vertical cracks & 12 & Dam’s elastic modulus zoning degradation for 40 years \\
4 & Contraction joints 0.25 GPa & 13 & Dam’s elastic modulus zoning degradation for 45 years \\
5 & Contraction joints 2 GPa & 14 & Dam’s elastic modulus zoning degradation for 10 years with all cracks \\
6 & Contraction joints 10 GPa & 15 & Dam’s elastic modulus zoning degradation for 20 years with all cracks \\
7 & Contraction joints and cracks & 16 & Dam’s elastic modulus zoning degradation for 30 years with all cracks \\
8 & Dam material zoning & - & Dam’s elastic modulus zoning degradation for 40 years with all cracks \\
9 & Dam’s elastic modulus zoning degradation for 10 years & - & Dam’s elastic modulus zoning degradation for 45 years with all cracks \\ \hline \hline \end{tabular} Notes: (I) vertical cracks in the table include selected longitudinal vertical cracks on the dam crest and vertical cracks on the upstream surface of the dam; (2) the modulus of elasticity of the contraction joints in case 7 and cases 14–18 is set to 0.25 GPa.  
\end{table} 
Table 6: Setup of cases considering the evolution of structural states and material properties. consistent with reference [42]. If the elastic modulus of contraction joints degrades over 45 years, as according to Equation (13), its elastic modulus only degrades to about 12.5 GPa, and the corresponding dynamic elastic modulus is about 18.8 GPa.