## 2 Materials and Methods
### Numerical Methods  
The numerical model of the cage in this study was established using the hydrodynamic analysis software OrcaFlex. aDeveloped by ORCINA in the United Kingdom in 1986, OrcaFlex has undergone numerous version updates and improvements over the years, establishing itself as a powerful tool in the field of marine engineering dynamics analysis. The software employs time-domain analysis methods to conduct static and dynamic analysis calculations of various marine structures under complex environmental loads. Its functionalities include mooring analysis, mooring fatigue, static and dynamic response of risers, and vortex-induced vibration (VIV) of risers. Especially when simulating and analyzing relatively small, slender structures, OrcaFlex exhibits unparalleled advantages  
Figure 2: Diagram of components and environmental load for the novel SPM net cage. thanks to its built-in units like 6 Dbouy and 3 Dbouy, as well as its fully coupled time-domain analysis approach. The components of the SPM-GDW cage system, including the floating collar, netting and ropes, and mooring lines, are slender rod-like structures with small scales relative to the wavelength of waves. In OrcaFlex, such objects can be simulated by 6 Dbouy and 3 Dbouy elements. The software employs the lumped mass method to discretize the slender members into a series of lumped mass points (nodes) and connecting springs (segments) (Figure 3). Both 6 Dbouy and 3 Dbouy represent lumped mass points, but the difference is that 6 Dbouy accounts for the transfer of bending moments at the mass point, while 3 Dbouy does not consider bending moment effects. In this model, the cage's floating collar qualifies as a \"rigid\" component compared to the netting ropes and mooring lines, and its bending moment forces cannot be neglected. Therefore, the 6 Dbouy element is used to simulate the floating collar. The netting ropes and mooring lines are simulated using the 3 Dbouy element. This approach ensures an accurate simulation of the dynamic behavior of the cage under wave and current loads. Under the action of wave and current loads, the structural dynamics balance equation for the cage system is as follows:  
\[[M]\big{[}\ddot{x}\big{]}+[C]\big{[}\dot{x}\big{]}+[K][x]=F \tag{1}\]  
where \([M]\) denotes the mass matrix, which includes the original mass and the added mass; \([C]\) is the damping matrix, which includes viscous damping in this study; \([K]\) is the stiffness matrix, encompassing hydrostatic stiffness and mooring stiffness; \([x]\) is the displacement matrix; \(\big{[}\dot{x}\big{]}\) is the corresponding velocity matrix; and \(\big{[}\ddot{x}\big{]}\) is the acceleration matrix. \(F\) represents the external forces acting on the cage, expressed as follows:  
\[F=f_{G}+f_{b}+f_{w}+f_{c} \tag{2}\]  
where \(f_{G}\) is the self-gravity force, which is calculated based on the static buoyancy; \(f_{b}\) is the buoyancy force, which can be determined by calculating the instantaneously submerged floating collar structure; and \(f_{w}\) and are \(f_{c}\) the wave force and current force, respectively, which are determined by the Morison equation. The expression of the Morison equation is as follows:  
\[f=\big{(}C_{m}Wa_{F}-C_{a}Wa_{B}\big{)}+\frac{1}{2}\rho C_{d}Av_{r}|v_{r}| \tag{3}\]  
where \((C_{m}Wa_{F}-C_{a}Wa_{B})\) is the inertia force term, while \(\frac{1}{2}\rho C_{d}Av_{r}|v_{r}|\) is the drag force term. In this equation, \(W\) denotes the mass of the Morison element, \(C_{m}\) is the inertia coefficient,  
Figure 3: Diagram of lumped mass model of slender in OrcaFlex. \(C_{a}\) is the added mass coefficient, \(a_{F}\) is the fluid acceleration, \(a_{B}\) is the solid acceleration, \(\rho\) is the fluid density, \(C_{d}\) is the drag coefficient, \(A\) is the drag projection area of the Morison element, and \(v_{r}\) is the velocity of the fluid relative to the solid.