## 4 Results and Discussion
### Force-Displacement Relationship  
To find the difference of the inherent nature between the MPM and the SPM mooring layouts, their force-displacement relationships for the DeepCwind semisubmersible wind turbine are further calculated based on an open-source Python wrapper of the mooring solver MAP++ package [33]. The Python wrapper is called later by our in-house routine developed for this purpose (see the part of \"Instructions for code acquisition\").  
The comparison between Figures 4 and 5 shows that the heave mode is the only mode that has not been affected by the change of the mooring system. When the platform has a surge or sway displacement, the mooring restoring load exerting on the platform increases more quickly if it is moored by the SPM system rather than the MPM. On the contrary, the roll-roll stiffness and the pitch-pitch stiffness of the SPM system are less than those of the MPM system. In particular, the yaw-yaw stiffness decreases to zero in the SPM layout. It means that the platform can rotate freely with respect to the \(z\)-axis no matter how big the yaw displacement. In addition, the platform displacements in roll and pitch lead to surge and sway restoring as well, which are much larger than those of the MPM system.  
Figure 4: Load-displacement relationships for the DeepCwind MPM system (recalculated based on the descriptions in Section 5.3 of [27]), restoring force/moment against displacement: (**a**) in surge; (**b**) in sway; (**c**) in heave; (**d**) in roll; (**e**) in pitch and (**f**) in yaw.  
The python wrapper also possesses a built-in feature that it can predict the linearized mooring stiffness matrix of a floating system. If the mooring compliance were assumed linear and the mooring inertia and damping were ignored [27], the total load upon the support platform from all-lines contribution would be  
\[F_{i}^{Lines}(q)=F_{i}^{Lines,0}-C_{ij}^{Lines}q_{j}, \tag{5}\]  
where \(F_{i}^{Lines,0}\) represents the pre-tension at the fairleads from the weight of the mooring lines not resting on the seafloor in water, \(C_{ij}^{Lines}\) represents the (_i,j_) component of the restoring matrix, and \(q_{j}\) is the _j_th platform DOF. It is calculated from the FAST still-water simulation that \(F_{i}^{Lines,0}\) remains the same with that in Equations (5)-(14) of Reference [27]. However, via the calculation by the python wrapper linking with the MAP++ DLL, the linearized mooring stiffness matrix in the undisplaced position is found to be  
Figure 5: Load-displacement relationships for the DeepCwind SPM system, restoring force/moment against displacement: (**a**) in surge; (**b**) in sway; (**c**) in heave; (**d**) in roll; (**e**) in pitch and (**f**) in yaw.  
\[C_{ij}^{Lims}=\left[\begin{array}{cccccc}7.07e4\ N/m&0&0&0&-1.41e6\ N/rad&0\\ 0&7.07e4\ N/m&0&1.41e6\ N/rad&0&0\\ 0&0&1.91e4\ N/m&0&0&0\\ 0&1.41e6\ Nm/m&0&6.62e7\ Nm/rad&0&0\\ -1.41e6\ Nm/m&0&0&0&6.62e7\ Nm/rad&0\\ 0&0&0&0&0&0\end{array}\right]. \tag{6}\]  
From the comparison between Equation (6) with Equations (5-15) of Reference [27], it is found that, by applying the SPM system instead of the MPM, the changes take place at the rotational-rotational diagonal terms and the surge-pitch, sway-roll, pitch-surge and roll-sway crossing terms.